Problem 64
Find the general integral of the linear equation…
Find the general integral of the linear equation $y'-\dfrac{2}{x+1}y=(x+1)^3$.
Explanation of Method
Linear in $y$ with $P(x)=-\dfrac{2}{x+1}$, $Q(x)=(x+1)^3$. Use the integrating factor $\mu(x)=(x+1)^{-2}$.
Step 1: Find the integrating factor
$\mu(x)=e^{\int -\frac{2}{x+1}dx}=e^{-2\ln|x+1|}=(x+1)^{-2}$.
Step 2: Multiply and integrate
Then $\dfrac{d}{dx}\!\left[\dfrac{y}{(x+1)^2}\right]=(x+1)$, so $\dfrac{y}{(x+1)^2}=\dfrac{(x+1)^2}{2}+C$.
Result
Final answer:
\[\boxed{\,y=\dfrac{(x+1)^4}{2}+C(x+1)^2,\qquad C\in\mathbb{R}}\]