${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}{\mathit{e}}^{\mathbf{x}}$Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by $\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{18}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{23}\mathbf{)}\mathbf{,}$or.$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{24}\mathbf{)}\mathbf{,}$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see $\mathbf{,}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{26}\mathbf{)}\mathbf{,}$andthe discussion after$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}{\mathit{e}}^{\mathbf{x}}$

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Given equation is$y^{\text{'}\text{'}}+y=2x{e}^x$

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.

The given differential equation is

$\begin{array}{l}{y}^{\text{'}\text{'}}+y=2x{e}^x\\ D^2+1=0\end{array}$

The auxiliary equation can be written as

$\begin{array}{c}{m}^2+1=0\\ m=\pm i\end{array}$

The complementary function can be written as

$\begin{array}{c}C.F=C_1\sin x+C_2\cos x\\ y=e^x(Ax+B)\\ y^{\text{'}}=e^x(Ax+B)+A{e}^x\\ y^{\text{'}\text{'}}=e^x(Ax+2A+B)e^x(Ax+2A+B)+e^x(Ax+B)+A{e}^x\\ =2x{e}^x\end{array}$

Solve the problem further

$e^x(2Ax+2B+2A)=2x{e}^x$

$\begin{array}{c}A=1,B=-1\\ P.I=e^x(x-1)\\ C.S=C.F+P.I\\ C.S=C_1\sin x+C_2\cos x+e^x(x-1)\end{array}$

The solution of the differential equation can be written as

$y\left(x\right)=C_1\sin x+C_2\cos x+e^x(x-1)$