Problem 62
Find the general integral of the linear equation…
Find the general integral of the linear equation $y'+\dfrac{y}{1+x}+x^2=0$.
Explanation of Method
Write in standard linear form $y'+\dfrac{1}{1+x}y=-x^2$ and solve using the integrating factor $\mu(x)=1+x$.
Step 1: Find the integrating factor
$\mu(x)=e^{\int \frac{dx}{1+x}}=e^{\ln|1+x|}=1+x$.
Step 2: Multiply and integrate
Multiplying gives $\dfrac{d}{dx}\big[(1+x)y\big]=-x^2(1+x)=-x^2-x^3$, so
\[(1+x)y=-\frac{x^3}{3}-\frac{x^4}{4}+C.\]
Result
Final answer:
\[\boxed{\,y=\dfrac{C-\frac{x^3}{3}-\frac{x^4}{4}}{1+x},\qquad C\in\mathbb{R}}\]