Equació diferencial lineal

In MATHEMATICS , a ((((( linear DIFFERENTIAL EQUATION ))))) is of the form ..

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$Ly=f\ ,$ ..

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where the DIFFERENTIAL OPERATOR (( L )) is a LINEAR OPERATOR , (( y )) is the unknown function (such as a function of time y(t)) , and the RIGHT HAND SIDE &fnof ; is a given function of the same nature as (( y )) (called the ((((( source term )))))). For a function dependent on time we may write the equation more expressively as ..

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$Ly(t)=f(t)\ ,$ ..

and , even more precisely by bracketing ..

$L[y(t)]=f(t)\ ,$ ..

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The linear operator (( L )) may be considered to be of the form . ..

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$L_{n}(y)\equiv {\frac {d^{n}y}{dt^{n}}}+A_{1}(t){\frac {d^{n-1}y}{dt^{n-1}}}+\cdots +..A_{n-1}(t){\frac {dy}{dt}}+A_{n}(t)y\ ,$ ..

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The linearity condition on (( L )) rules out operations such as taking the square of the DERIVATIVE of (( y ))  ; but permits , for example , taking the second derivative of (( y )) . .. It is convenient to rewrite this equation in an operator form ..

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$L_{n}(y)\equiv \left[\ ,D^{n}+A_{1}(t)D^{n-1}+\cdots +A_{n-1}(t)D+A_{n}(t)\right]y$ ..

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where (( D )) is the differential operator (( d/dt )) (i.e. (( Dy = y' )) , (( D )) ²(( y = y" ,... )) ) , and the (( A_n )) are given functions. ..

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Such an equation is said to have ((((( order ))))) (( n )) , the index of the highest derivative of (( y )) that is involved. ..

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A typical simple example is the linear differential equation used to model radioactive decay . . Let N(t) denote the number of radioactive atoms in some sample of material (such as a portion of the cloth of the SHROUD OF TURIN . ) at time t. Then for some constant k > 0 , .. the number of radioactive atoms which decay can be modelled by ..

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If (( y )) is assumed to be a function of only one variable , one speaks about an ORDINARY DIFFERENTIAL EQUATION , else the derivatives and their coefficients must be understood as (CONTRACTED) vectors , matrices or TENSORS of higher rank , and we have a (linear) PARTIAL DIFFERENTIAL EQUATION. ..

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The case where &fnof ; = 0 is called a ((((( homogeneous equation ))))) and its solutions are called ((((( complementary functions ))))). It is particularly important to the solution of the general case , since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called (( particular integral and complementary function )) ). When the (( A_i )) are numbers , the equation is said to have (( CONSTANT COEFFICIENTS )) . ..

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(==) Homogeneous equations with constant coefficients (==) ..

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The first method of solving linear ordinary differential equations with constant coefficients is due to EULER , who realized that solutions have the form $e^{zx}$ , for possibly-complex values of $z$. The exponential function is one of the few functions that keep its shape even after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero , the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus , to solve ..

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$y^{(n)}+A_{1}y^{(n-1)}+\cdots +A_{n}y=0$ ..

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we set $y=e^{zx}$ , leading to ..

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$z^{n}e^{zx}+A_{1}z^{n-1}e^{zx}+\cdots +A_{n}e^{zx}=0.$ ..

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Division by (( e )) ^{&nbsp ;(( zx ))} gives the (( n )) th-order polynomial ..

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$F(z)=z^{n}+A_{1}z^{n-1}+\cdots +A_{n}=0.\ ,$ ..

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This algebraic equation (( F )) (((( z )))) = 0 , is the ((((( characteristic equation ))))) considered later by MONGE and CAUCHY. ..

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Formally , the terms ..

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$y^{(k)}\quad \quad (k=1,2,\dots ,n).$ ..

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of the original differential equation are replaced by (( z )) ^{(( k ))} . ROOT-FINDING ALGORITHM#FINDING ROOTS OF POLYNOMIALS .3. SOLVING .3. Solving]] the polynomial gives (( n )) values of (( z )) , (( z )) ₁ ,&nbsp ;... ,&nbsp ;(( z )) _{(( n ))} . Substitution of any of those values for (( z )) into (( e )) ^{&nbsp ;(( zx ))} gives a solution (( e )) ^{&nbsp ;(( z )) _{(( i ))} (( x ))} . Since homogeneous linear differential equations obey the SUPERPOSITION PRINCIPLE , any LINEAR COMBINATION of these functions also satisfies the differential equation. ..

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When these roots are all DISTINCT , we have (( n )) distinct solutions to the differential equation. It can be shown that these are LINEARLY INDEPENDENT , by applying the VANDERMONDE DETERMINANT , and together they form a BASIS of the space of all solutions of the differential equation. ..

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{{ ExampleSidebar .3. 35 % .3. ..

$y((((('-2y)))))+2y''-2y'+y=0\ ,$ ..

has the characteristic equation ..

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$z^{4}-2z^{3}+2z^{2}-2z+1=0.\ ,$ ..

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This has zeroes , (( i )) , &minus ;(( i )) , and 1 (multiplicity 2). The solution basis is then ..

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$e^{ix},\ ,e^{-ix},\ ,e^{x},\ ,xe^{x}\ ,.$ ..

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This corresponds to the real-valued solution basis ..

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$\cos x,\ ,\sin x,\ ,e^{x},\ ,xe^{x}\ ,.$}} ..

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The preceding gave a solution for the case when all zeros are distinct , that is , each has MULTIPLICITY#MULTIPLICITY OF A ROOT OF A POLYNOMIAL .3. MULTIPLICITY .3. multiplicity]] 1. For the general case , if (( z )) is a (possibly complex) ZERO (or root) of (( F )) (((( z )))) having multiplicity (( m )) , then , for No s'ha pogut entendre (MathML amb SVG o PNG alternatiu (recomanat per a navegadors moderns i eines d'accessibilitat): Resposta invàlida («Math extension cannot connect to Restbase.») del servidor «http://localhost:6011/ca.wikipedia.org/v1/»:): {\displaystyle k\in\{0 ,1 ,\dots ,m-1\} \ ,} , $y=x^{k}e^{zx}\ ,$ is a solution of the Ode. Applying this to all roots gives a collection of (( n )) distinct and linearly independent functions , where (( n )) is the degree of (( F )) (((( z )))) . As before , these functions make up a basis of the solution space. ..

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If the coefficients (( A_i )) of the differential equation are real , then real-valued solutions are generally preferable. Since non-real roots (( z )) then come in CONJUGATE pairs , so do their corresponding basis functions {{ nowrap .3. (( x )) ^{(( k ))} e^{(( zx ))} }} , and the desired result is obtained by replacing each pair with their real-valued LINEAR COMBINATIONS RE(((( Y )))) and IM(((( Y )))) , where (( y )) is one of the pair. ..

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A case that involves complex roots can be solved with the aid of EULER'S FORMULA. ..

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(===) Examples (===) ..

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Given $y((-4y'+5y=0\ ,$. The characteristic equation is $z^{2}-4z+5=0\ ,$ which has zeroes 2+ )) i(( and 2− )) i(( . Thus the solution basis $\{y_{1},y_{2}\}$ is $\{e^{(2+i)x},e^{(2-i)x}\}\ ,$. Now )) y is a solution IF AND ONLY IF $y=c_{1}y_{1}+c_{2}y_{2}\ ,$ for No s'ha pogut entendre (MathML amb SVG o PNG alternatiu (recomanat per a navegadors moderns i eines d'accessibilitat): Resposta invàlida («Math extension cannot connect to Restbase.») del servidor «http://localhost:6011/ca.wikipedia.org/v1/»:): {\displaystyle c_1 ,c_2\in\mathbb C} . ..

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Because the coefficients are real , ..

• we are likely not interested in the complex solutions ..
• our basis elements are mutual conjugates ..

The linear combinations ..

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$u_{2}={\mbox{Im}}(y_{1})={\frac {y_{1}-y_{2}}{2i}}=e^{2x}\sin(x)\ ,$ ..

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will give us a real basis in $\{u_{1},u_{2}\}$. ..

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(====) Simple harmonic oscillator (====) ..

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The second order differential equation ..

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$D^{2}y=-k^{2}y,$ ..

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which represents a simple HARMONIC OSCILLATOR , can be restated as ..

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$(D^{2}+k^{2})y=0.$ ..

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The expression in parenthesis can be factored out , yielding ..

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$(D+ik)(D-ik)y=0,$ ..

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which has a pair of linearly independent solutions , one for ..

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$(D-ik)y=0$ ..

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and another for ..

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$(D+ik)y=0.$ ..

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The solutions are , respectively , ..

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$y_{0}=A_{0}e^{ikx}$ ..

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and ..

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$y_{1}=A_{1}e^{-ikx}.$ ..

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These solutions provide a basis for the two-dimensional "SOLUTION SPACE" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular , the following solutions can be constructed ..

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$y_{0'}={A_{0}e^{ikx}+A_{1}e^{-ikx} \over 2}=C_{0}\cos(kx)$ ..

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and ..

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$y_{1'}={A_{0}e^{ikx}-A_{1}e^{-ikx} \over 2i}=C_{1}\sin(kx).$ ..

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These last two trigonometric solutions are linearly independent , so they can serve as another basis for the solution space , yielding the following general solution: ..

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$y_{H}=C_{0}\cos(kx)+C_{1}\sin(kx).$ ..

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(====) Damped harmonic oscillator (====) .. Given the equation for the damped HARMONIC OSCILLATOR: ..

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$\left(D^{2}+{b \over m}D+\omega _{0}^{2}\right)y=0,$ ..

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the expression in parentheses can be factored out: first obtain the characteristic equation by replacing (( D )) with &lambda ;. This equation must be satisfied for all (( y )) , thus: ..

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$\lambda ^{2}+{b \over m}\lambda +\omega _{0}^{2}=0.$ ..

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Solve using the QUADRATIC FORMULA: ..

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$\lambda ={-b/m\pm {\sqrt {b^{2}/m^{2}-4\omega _{0}^{2}}} \over 2}.$ ..

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Use these data to factor out the original differential equation: ..

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$\left(D+{b \over 2m}-{\sqrt {{b^{2} \over 4m^{2}}-\omega _{0}^{2}}}\right)\left(D+{b \over 2m}+{\sqrt {{b^{2} \over 4m^{2}}-\omega _{0}^{2}}}\right)y=0.$ ..

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This implies a pair of solutions , one corresponding to ..

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$\left(D+{b \over 2m}-{\sqrt {{b^{2} \over 4m^{2}}-\omega _{0}^{2}}}\right)y=0$ ..

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and another to ..

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$\left(D+{b \over 2m}+{\sqrt {{b^{2} \over 4m^{2}}-\omega _{0}^{2}}}\right)y=0$ ..

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The solutions are , respectively , ..

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$y_{0}=A_{0}e^{-\omega x+{\sqrt {\omega ^{2}-\omega _{0}^{2}}}x}=A_{0}e^{-\omega x}e^{{\sqrt {\omega ^{2}-\omega _{0}^{2}}}x}$ ..

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and ..

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$y_{1}=A_{1}e^{-\omega x-{\sqrt {\omega ^{2}-\omega _{0}^{2}}}x}=A_{1}e^{-\omega x}e^{-{\sqrt {\omega ^{2}-\omega _{0}^{2}}}x}$ ..

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where &omega ; = (( b )) / 2(( m )) . From this linearly independent pair of solutions can be constructed another linearly independent pair which thus serve as a basis for the two-dimensional solution space: ..

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$y_{H}(A_{0},A_{1})(x)=\left(A_{0}\sinh {\sqrt {\omega ^{2}-\omega _{0}^{2}}}x+A_{1}\cosh {\sqrt {\omega ^{2}-\omega _{0}^{2}}}x\right)e^{-\omega x}.$ ..

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However , if .3. &omega ; .3. < .3. &omega ;₀ .3. then it is preferable to get rid of the consequential imaginaries , expressing the general solution as ..

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$y_{H}(A_{0},A_{1})(x)=\left(A_{0}\sin {\sqrt {\omega _{0}^{2}-\omega ^{2}}}x+A_{1}\cos {\sqrt {\omega _{0}^{2}-\omega ^{2}}}x\right)e^{-\omega x}.$ ..

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This latter solution corresponds to the underdamped case , whereas the former one corresponds to the overdamped case: the solutions for the underdamped case OSCILLATE whereas the solutions for the overdamped case do not. ..

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(==) Nonhomogeneous equation with constant coefficients (==) ..

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To obtain the solution to the ((((( non-homogeneous equation ))))) (sometimes called ((((( inhomogeneous equation )))))) , find a particular solution (( y )) _{(( P ))} (((( x )))) by either the METHOD OF UNDETERMINED COEFFICIENTS or the METHOD OF VARIATION OF PARAMETERS ; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution. ..

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Suppose we face ..

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${\frac {d^{n}y(x)}{dx^{n}}}+A_{1}{\frac {d^{n-1}y(x)}{dx^{n-1}}}+\cdots +A_{n}y(x)=f(x).$ ..

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For later convenience , define the characteristic polynomial ..

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$P(v)=v^{n}+A_{1}v^{n-1}+\cdots +A_{n}.$ ..

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We find the solution basis $\{y_{1}(x),y_{2}(x),\ldots ,y_{n}(x)\}$ as in the homogeneous (((( f(x)=0 )))) case. We now seek a ((((( particular solution ))))) (( y_p(x) )) by the ((((( variation of parameters ))))) method. Let the coefficients of the linear combination be functions of (( x )) : ..

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$y_{p}(x)=u_{1}(x)y_{1}(x)+u_{2}(x)y_{2}(x)+\cdots +u_{n}(x)y_{n}(x).$ ..

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For ease of notation we will drop the dependency on (( x )) (i.e. the various ((( x ))) ). Using the "operator" notation $D=d/dx$ and a broad-minded use of notation , the Ode in question is $P(D)y=f$ ; so ..

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$f=P(D)y_{p}=P(D)(u_{1}y_{1})+P(D)(u_{2}y_{2})+\cdots +P(D)(u_{n}y_{n}).$ ..

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With the constraints ..

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$0=u'_{1}y_{1}+u'_{2}y_{2}+\cdots +u'_{n}y_{n}$ ..

$0=u'_{1}y'_{1}+u'_{2}y'_{2}+\cdots +u'_{n}y'_{n}$ ..

$\cdots$ ..

$0=u'_{1}y_{1}^{(n-2)}+u'_{2}y_{2}^{(n-2)}+\cdots +u'_{n}y_{n}^{(n-2)}$ ..

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the parameters commute out , with a little "dirt": ..

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$f=u_{1}P(D)y_{1}+u_{2}P(D)y_{2}+\cdots +u_{n}P(D)y_{n}+u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.$ ..

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But $P(D)y_{j}=0$ , therefore ..

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$f=u'_{1}y_{1}^{(n-1)}+u'_{2}y_{2}^{(n-1)}+\cdots +u'_{n}y_{n}^{(n-1)}.$ ..

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This , with the constraints , gives a linear system in the $u'_{j}$. This much can always be solved ; in fact , combining CRAMER'S RULE with the WRONSKIAN , ..

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$u'_{j}=(-1)^{n+j}{\frac {W(y_{1},\ldots ,y_{j-1},y_{j+1}\ldots ,y_{n})_{0 \choose f}}{W(y_{1},y_{2},\ldots ,y_{n})}}.$ ..

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The rest is a matter of integrating $u'_{j}.$ ..

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The particular solution is not unique ; $y_{p}+c_{1}y_{1}+\cdots +c_{n}y_{n}$ also satisfies the Ode for any set of constants (( c_j )) . ..

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(===) Example (===) .. Suppose $y''-4y'+5y=sin(kx)$. We take the solution basis found above $\{e^{(2+i)x},e^{(2-i)x}\}$. ..

{ .3. ..

.3. - ..
.3. $W\ ,$ ..
.3. $={\begin{vmatrix}e^{(2+i)x}&e^{(2-i)x}\\(2+i)e^{(2+i)x}&(2-i)e^{(2-i)x}\end{vmatrix}}$ ..
.3. - ..
.3.  ..
.3. $=e^{4x}{\begin{vmatrix}1&1\\2+i&2-i\end{vmatrix}}$ ..
.3. - ..
.3.  ..
.3. $=-2ie^{4x}\ ,$ ..
.3. } ..
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{ .3. ..

.3. - ..
.3. $u'_{1}\ ,$ ..
.3. $={\frac {1}{W}}{\begin{vmatrix}0&e^{(2-i)x}\\\sin(kx)&(2-i)e^{(2-i)x}\end{vmatrix}}$ ..
.3. - ..
.3.  ..
.3. $=-{\frac {i}{2}}\sin(kx)e^{(-2-i)x}$ ..
.3. } ..
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{ .3. ..

.3. - ..
.3. $u'_{2}\ ,$ ..
.3. $={\frac {1}{W}}{\begin{vmatrix}e^{(2+i)x}&0\\(2+i)e^{(2+i)x}&\sin(kx)\end{vmatrix}}$ ..
.3. - ..
.3.  ..
.3. $={\frac {i}{2}}\sin(kx)e^{(-2+i)x}.$ ..
.3. } ..
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Using the LIST OF INTEGRALS OF EXPONENTIAL FUNCTIONS ..

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{ .3. ..

.3. - ..
.3. $u_{1}\ ,$ ..
.3. $=-{\frac {i}{2}}\int \sin(kx)e^{(-2-i)x}\ ,dx$ ..
.3. - ..
.3.  ..
.3. $={\frac {ie^{(-2-i)x}}{2(3+4i+k^{2})}}\left((2+i)\sin(kx)+k\cos(kx)\right)$ ..
.3. } ..
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{ .3. ..

.3. - ..
.3. $u_{2}\ ,$ ..
.3. $={\frac {i}{2}}\int \sin(kx)e^{(-2+i)x}\ ,dx$ ..
.3. - ..
.3.  ..
.3. $={\frac {ie^{(i-2)x}}{2(3-4i+k^{2})}}\left((i-2)\sin(kx)-k\cos(kx)\right).$ ..
.3. } ..
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And so ..

{ .3. ..

.3. - ..
.3. $y_{p}\ ,$ ..
.3. $={\frac {i}{2(3+4i+k^{2})}}\left((2+i)\sin(kx)+k\cos(kx)\right)..+{\frac {i}{2(3-4i+k^{2})}}\left((i-2)\sin(kx)-k\cos(kx)\right)$ ..
.3. - ..
.3.  ..
.3. $={\frac {(5-k^{2})\sin(kx)+4k\cos(kx)}{(3+k^{2})^{2}+16}}.$ ..
.3. } ..

(Notice that (( u )) ₁ and (( u )) ₂ had factors that canceled (( y )) ₁ and (( y )) ₂ ; that is typical.) ..

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For interest's sake , this Ode has a physical interpretation as a driven damped HARMONIC OSCILLATOR ; (( y_p )) represents the steady state , and $c_{1}y_{1}+c_{2}y_{2}$ is the transient. ..

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== Equation with variable coefficients== ..

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A linear Ode of order (( n )) with variable coefficients has the general form ..

$p_{n}(x)y^{(n)}(x)+p_{n-1}(x)y^{(n-1)}(x)+\cdots +p_{0}(x)y(x)=r(x).$ ..

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(===) Examples (===) ..

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A simple example is the CAUCHY–EULER EQUATION often used in engineering ..

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$x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0.$ ..

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(==) First order equation (==) .. {{ ExampleSidebar .3. 35 % .3. Solve the equation ..

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$y'(x)+3y(x)=2\ ,$ ..

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with the initial condition ..

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$y\left(0\right)=2.\ ,$ ..

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Using the general solution method: ..

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$y=e^{-3x}\left(\int 2e^{3x}\ ,dx+\kappa \right).\ ,$ ..

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The indefinite integral is solved to give: ..

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$y=e^{-3x}\left(2/3e^{3x}+\kappa \right).\ ,$ ..

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Then we can reduce to: ..

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where (( &kappa ; )) is 4/3 from the initial condition.}} .. A linear Ode of order 1 with variable coefficients has the general form ..

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$Dy(x)+f(x)y(x)=g(x).$ ..

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Equations of this form can be solved by multiplying the INTEGRATING FACTOR ..

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$e^{\int f(x)\ ,dx}$ ..

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throughout to obtain ..

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$Dy(x)e^{\int f(x)\ ,dx}+f(x)y(x)e^{\int f(x)\ ,dx}=g(x)e^{\int f(x)\ ,dx},$ ..

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which simplifies due to the PRODUCT RULE to ..

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$D(y(x)e^{\int f(x)\ ,dx})=g(x)e^{\int f(x)\ ,dx}$ ..

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which , on integrating both sides , yields ..

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$y(x)e^{\int f(x)\ ,dx}=\int g(x)e^{\int f(x)\ ,dx}\ ,dx+c~,$ ..

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$y(x)={\int g(x)e^{\int f(x)\ ,dx}\ ,dx+c \over e^{\int f(x)\ ,dx}}~.$ ..

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In other words: The solution of a first-order linear Ode ..

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$y'(x)+f(x)y(x)=g(x),$ ..

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with coefficients that may or may not vary with (( x )) , is: ..

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$y=e^{-a(x)}\left(\int g(x)e^{a(x)}\ ,dx+\kappa \right)$ ..

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where (( $\kappa$ )) is the constant of integration , and ..

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$a(x)=\int {f(x)\ ,dx}.$ ..

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(===) Examples (===) .. Consider a first order differential equation with CONSTANT COEFFICIENTS: ..

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${\frac {dy}{dx}}+by=1.$ ..

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This equation is particularly relevant to first order systems such as RC CIRCUITS and MASS-DAMPER systems. ..

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In this case , (( p )) (((( x )))) = b , (( r )) (((( x )))) = 1. ..

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Hence its solution is ..

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$y(x)=e^{-bx}\left(e^{bx}/b+C\right)=1/b+Ce^{-bx}.$ ..

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(==) See also (==) ..

• CONTINUOUS-REPAYMENT_MORTGAGE#ORDINARY_TIME_DIFFERENTIAL_EQUATION .3. CONTINUOUS-REPAYMENT MORTGAGE .3. Continuous-repayment mortgage]] ..
• FOURIER TRANSFORM ..
• LAPLACE TRANSFORM ..

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(==) Notes (==) .. Plantilla:Reflist .3. 2 ..

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(==) References (==) ..

• {{ Citation ..

 .3.  author = Birkhoff , Garret and Rota , Gian-Carlo ..
 .3.  year = 1978 ..
 .3.  title = Ordinary Differential Equations ..
 .3.  isbn = 0-471-07411-X ..
 .3.  publisher = John Wiley and Sons , Inc. ..
 .3.  location = New York ..
 .3.  oclc = ..

}} ..

• {{ Citation ..

 .3.  author = Gershenfeld , Neil ..
 .3.  year = 1999 ..
 .3.  title =The Nature of Mathematical Modeling ..
 .3.  isbn = 978-0521-570954 ..
 .3.  publisher = Cambridge University Press ..
 .3.  location = Cambridge , Uk. ..
 .3.  oclc = ..

}} ..

• {{ Citation ..

 .3.  author = Robinson , James C. ..
 .3.  year = 2004 ..
 .3.  title = An Introduction to Ordinary Differential Equations ..
 .3.  isbn = 0-521-826500 ..
 .3.  publisher = Cambridge University Press ..
 .3.  location = Cambridge , Uk. ..
 .3.  oclc = ..

}} ..

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paraulesenllacos ..

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MATHEMATICS ..

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DIFFERENTIAL EQUATION ..

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DIFFERENTIAL OPERATOR ..

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LINEAR OPERATOR ..

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RIGHT HAND SIDE ..

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DERIVATIVE ..

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SHROUD OF TURIN ..

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ORDINARY DIFFERENTIAL EQUATION ..

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CONTRACTED ..

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TENSORS ..

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PARTIAL DIFFERENTIAL EQUATION ..

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CONSTANT COEFFICIENTS ..

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EULER ..

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MONGE ..
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CAUCHY ..

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ROOT-FINDING ALGORITHM#FINDING ROOTS OF POLYNOMIALS .3. SOLVING ..

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SUPERPOSITION PRINCIPLE ..

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LINEAR COMBINATION ..

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DISTINCT ..

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LINEARLY INDEPENDENT ..

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VANDERMONDE DETERMINANT ..

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BASIS ..

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MULTIPLICITY#MULTIPLICITY OF A ROOT OF A POLYNOMIAL .3. MULTIPLICITY ..

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ZERO ..

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CONJUGATE ..

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LINEAR COMBINATIONS ..

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RE(((( Y )))) ..

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IM(((( Y )))) ..

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EULER'S FORMULA ..

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IF AND ONLY IF ..

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HARMONIC OSCILLATOR ..

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SOLUTION SPACE ..

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HARMONIC OSCILLATOR ..

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QUADRATIC FORMULA ..

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OSCILLATE ..

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METHOD OF UNDETERMINED COEFFICIENTS ..

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METHOD OF VARIATION OF PARAMETERS ..

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CRAMER'S RULE ..

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WRONSKIAN ..

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LIST OF INTEGRALS OF EXPONENTIAL FUNCTIONS ..

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HARMONIC OSCILLATOR ..

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CAUCHY–EULER EQUATION ..

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INTEGRATING FACTOR ..

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PRODUCT RULE ..

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CONSTANT COEFFICIENTS ..

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RC CIRCUITS ..

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MASS-DAMPER ..

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CONTINUOUS-REPAYMENT_MORTGAGE#ORDINARY_TIME_DIFFERENTIAL_EQUATION .3. CONTINUOUS-REPAYMENT MORTGAGE ..

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FOURIER TRANSFORM ..

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LAPLACE TRANSFORM ..