Problem 89
Find the general integral of the Bernoulli equation…
Find the general integral of the Bernoulli equation $y'+\dfrac{y}{x+1}+y^2=0$.
Explanation of Method
With $n=2$, substitute $v=1/y$ to reduce to a linear equation for $v(x)$.
Step 1: Substitute $v=1/y$
Dividing by $y^2$ and using $v'=-y'/y^2$: $-v'+\dfrac{v}{x+1}=-1$, i.e. $v'-\dfrac{v}{x+1}=1$.
Step 2: Solve the linear equation
With $\mu(x)=\dfrac{1}{x+1}$: $\dfrac{d}{dx}\!\left(\dfrac{v}{x+1}\right)=\dfrac{1}{x+1}$, so $\dfrac{v}{x+1}=\ln|x+1|+C$.
Result
Final answer:
\[\boxed{\,\frac{1}{y}=(x+1)\big[\ln|x+1|+C\big],\qquad C\in\mathbb{R}}\]