Question

Solve the given initial value problem. Give the largest interval lover which the general solution is defined.

$\mathbf{(}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{l}\mathit{n}\mathit{x}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{10}$

Short answer

The solution for the given initial valueis$y=\frac{x}{x+1}\ln x-\frac{x}{x+1}+\frac{21}{x+1}$

Step-by-step solution

The given equation in the standard form and determine the integrated factor

Dividing the given differential equation by (x+1), we get

$\frac{dy}{dx}+\frac{y}{x+1}=\frac{\ln x}{x+1}$

which is in the standard form with $P\left(x\right)=\frac{1}{x+1}andf\left(x\right)=\frac{{\ln }^x}{x+1}$.

Note that p and f are continuous on .

Hence, an integrating factor is

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

$$\begin{gathered} =|x+1| \\ =x+1 \end{gathered}$$

Since, x+1>0 on role="math" localid="1664169970239" $(0,\infty ),|x+1|$can be replaced by $(x+1)$.