Solving Linear Differential Equations
Solving Linear Differential Equations:

Homogeneous Linear Differential Equations:
 Characteristic Equation Method:
 Assume the solution of the form $y={e}^{rx}$.
 Substitute this solution into the homogeneous equation to form a characteristic equation using derivatives.
 Solve the characteristic equation to find roots ${r}_{1},{r}_{2},\dots ,{r}_{n}$.
 The general solution is $y={C}_{1}{e}^{{r}_{1}x}+{C}_{2}{e}^{{r}_{2}x}+\cdots +{C}_{n}{e}^{{r}_{n}x}$, where ${C}_{1},{C}_{2},\dots ,{C}_{n}$are constants.
 Characteristic Equation Method:

NonHomogeneous Linear Differential Equations:

Method of Undetermined Coefficients:
 For the nonhomogeneous equation ${a}_{n}(x)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n1}(x)\frac{{d}^{n1}y}{d{x}^{n1}}+\cdots +{a}_{1}(x)\frac{dy}{dx}+{a}_{0}(x)y=g(x)$:
 Solve the associated homogeneous equation to find the complementary function (CF).
 Guess a form for the particular solution (PS) based on $g(x)$ and its derivatives.
 Substitute the guessed form into the nonhomogeneous equation, solve for undetermined coefficients, and add the CF to get the general solution.

Variation of Parameters:
 Applicable when the nonhomogeneous term $g(x)$ is expressed as a function.
 Assume the form of a particular solution ${y}_{p}(x)$ with undetermined coefficients involving the complementary function.
 Determine the variation parameters involving integrals of functions derived from the homogeneous solution.
 Combine the CF with the PS obtained through variation of parameters to form the general solution.


Example: Solving a Linear Differential Equation by Undetermined Coefficients:
Consider the equation ${y}^{\mathrm{\prime}\mathrm{\prime}}4y=3{x}^{2}+2{e}^{x}$.
a. Solve the Associated Homogeneous Equation:
 The associated homogeneous equation is ${y}^{\mathrm{\prime}\mathrm{\prime}}4y=0$, leading to the CF: ${y}_{CF}={C}_{1}{e}^{2x}+{C}_{2}{e}^{2x}$.
b. Guess a Form for the Particular Solution (PS):
 Based on $3{x}^{2}$ and $2{e}^{x}$, assume ${y}_{p}(x)=A{x}^{2}+B{e}^{x}$, where $A$ and $B$ are undetermined coefficients.
c. Substitute the Guessed Form into the NonHomogeneous Equation:
 Find ${y}^{\mathrm{\prime}}$ and ${y}^{\mathrm{\prime}\mathrm{\prime}}$for ${y}_{p}(x)$ and substitute into the nonhomogeneous equation.
 Equate coefficients of similar terms on both sides to solve for $A$ and $B$.
d. Combine the CF with the PS to Form the General Solution:
 The general solution is $y={C}_{1}{e}^{2x}+{C}_{2}{e}^{2x}+A{x}^{2}+B{e}^{x}$, where $A$ and $B$ are the determined coefficients.

Example of Solving a Linear Differential Equation:
Consider the differential equation: ${y}^{\mathrm{\prime}\mathrm{\prime}}4{y}^{\mathrm{\prime}}+4y=3{e}^{2x}$.
a. Homogeneous Solution (CF):
 The associated homogeneous equation is ${y}^{\mathrm{\prime}\mathrm{\prime}}4{y}^{\mathrm{\prime}}+4y=0$.
 Characteristic equation: ${r}^{2}4r+4=0$ gives a repeated root $r=2$.
 The CF is ${y}_{CF}=({C}_{1}+{C}_{2}x){e}^{2x}$.
b. Particular Solution (PS) by Method of Undetermined Coefficients:
 Assume a particular solution of the form ${y}_{PS}=A{e}^{2x}$ for the nonhomogeneous equation.
 Substitute into the equation and solve for $A$: $3{e}^{2x}=3A{e}^{2x}$. Therefore, $A=1$.
 The PS is ${y}_{PS}={e}^{2x}$.
c. General Solution:
 The general solution is the sum of CF and PS: $y={y}_{CF}+{y}_{PS}=({C}_{1}+{C}_{2}x){e}^{2x}+{e}^{2x}$.