Question

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$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{24}\mathbf{)}\mathbf{,}$ .Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$\mathbf{(}\mathbf{6}\mathbf{.26}\mathbf{)}$ andthe discussion after].

$\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}\mathit{x}$

Short answer

The general solution given by differential equation is$y\left(x\right)=C_1+C_2{e}^{-\frac{x}{2}}+x^2-4x$

Step-by-step solution

Given data. 

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

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.2y''+y'=2x

The given differential equation is

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

The auxiliary equation can be written as

$\begin{array}{c}2m^2+m=0\\ m(2m+1)=0\\ m=0,-\frac{1}{2}\end{array}$

Solve complementary function and particular integral.

$C.F=C_1+C_2{e}^{-\frac{x}{2}}$

$\begin{array}{c}P.I=\frac{1}{2D^2+D}2x\\ \Rightarrow \frac{1}{D(2D+1)}2x\\ \Rightarrow \frac{1}{D}{\left(1+2D\right)}^{-1}2x\\ \Rightarrow \frac{1}{D}(1-2D)2x\end{array}$

Solve the equation further

$\begin{array}{c}\Rightarrow \frac{1}{D}(2x-4)\\ \Rightarrow x^2-4x\\ P.I=x^2-4x\end{array}$

Solution of the differential equation can be written as,

$y\left(x\right)=C_1+C_2{e}^{-\frac{x}{2}}+x^2-4x$