Question

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by $(\mathbf{6}.\mathbf{18}),(\mathbf{6}.\mathbf{23}),$or.$(\mathbf{6}.24),$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$(\mathbf{6}.26),$and the discussionafter$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].

$\mathbf{(}\mathit{D}\mathbf{-}\mathbf{3}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{16}{\mathit{x}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{x}}$

Short answer

The general solution given by differential equation is$y\left(x\right)=(C_1+C_{2}x+2x^3)e^{3x}$

Step-by-step solution

Given data. 

Given equation is.$(D-3)(D+1)y=16x^2{e}^{-x}$

General solution of differential equation.

A general solution to the nth order differential equation is one that incorporates a significant number of arbitrary constants. If one uses the variable approach to solve a first-order differential equation, one must insert an arbitrary constant as soon as integration is completed.

Find the general solution of given differential equation.(D−3)(D+1)y=16x2e−x

Solve the differential equation

$(D-3)(D+1)y=16x^2{e}^{-x}$

The auxiliary equation can be written as

$\begin{array}{l}\Rightarrow m^2-2m-3=0\\ \Rightarrow m^2-3m+m-3=0\\ \Rightarrow m(m-3)+1(m-3)\end{array}$

The roots of the equation are

$\begin{array}{l}\Rightarrow m=-1,3\\ C.F=C_1{e}^{-x}+C_2{e}^{3x}\\ y=x{e}^x(A{x}^2+Bx+C)\\ y^{\text{'}}=-e^{-x}(A{x}^3+B{x}^2+Cx)+e^{-x}(3A{x}^2+2Bx+C)\end{array}$

Differentiate again,

$y^{\text{'}\text{'}}=e^{-x}(C(x-2)+Ax(6-6x+x^2)+B(2-4x+x^2)$

Rearrange and get

$\begin{array}{c}B-2C-4Bx+3Ax-6A{x}^2=8x^2\\ (D-3)(D+1)y=16x^2{e}^{-x}\end{array}$

Solve the problem further

$\begin{array}{l}A=\frac{-4}{3},B=-1,C=-\frac{1}{2}\\ P.I=e^{-x}(-\frac{4}{3}{x}^3-x^2-\frac{x}{2})\\ \text{C.}S=C.F+P.I\end{array}$

Hence,

$\begin{array}{l}\text{C.S}=C_1{e}^{-x}+C_2{e}^{3x}+e^{-x}(-\frac{4}{3}{x}^3-x^2-\frac{x}{2})\\ \text{C.}S=(C_1+C_{2}x+2x^3)e^{3x}\end{array}$

Hence, the solution of the differential equation can be written as

$y\left(x\right)=(C_1+C_{2}x+2x^3)e^{3x}$