Question

In problems 1-24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution.

$(x+1)\frac{dy}{dx}+(x+2)y=2x{e}^{-x}$

Short answer

The general solution is$e^x(x+1)y=x^2+c,\left(-1,\infty \right)$

Step-by-step solution

Concept of Linear Differential Equation

$\frac{dy}{dx}+py=q$, where p and q can be constants or variables.

Convert the given differential equation into the standard form of linear equation.

Given differential equation

$$\begin{gathered} (x+1)\frac{dy}{dx}+(x+2)y=2x{e}^{-x} \\ \frac{dy}{dx}+\frac{(x+2)}{(x+1)}y=\frac{2x{e}^{-x}}{x+1} \end{gathered}$$

The function $P\left(x\right)=\frac{(x+2)}{(x+1)}$ is continuous on $(0,\infty )$

Find the integrating factor 

Now the integrating factor of the equation is

$e^{\int \frac{(x+2)}{(x+1)}dx}=e^{\int 1+\frac{1}{x+1}}=e^{x+\ln |x+1|}=e^x(x+1)$

Find the general solution

$$\begin{gathered} {}^{\int \frac{(x+2)}{(x+1)}dx}=e^{\int 1+\frac{1}{x+1}}=e^{x+\ln |x+1|}=e^x(x+1) \\ {e}^x(x+1)\frac{dy}{dx}+e^x(x+1)\frac{(x+2)}{(x+1)}y=2x \\ \frac{d}{dx}\left[e^x(x+1)y\right]=2x \end{gathered}$$

Integrating Both Sides

$$\begin{gathered} \frac{d}{dx}\left[e^x(x+1)y\right]=2x \\ {e}^x(x+1)y=x^2+c \end{gathered}$$