If y=y(x) solves (5+e^x)^2dydx (schematic: (5+e^x),dy/(2+y)=-e^x,dx), y(0)=1, then a value…

Question

If $y=y(x)$ solves $\big(5+e^x\big)^2\dfrac{dy}{dx}\ (	ext{schematic: }(5+e^x)\,dy/(2+y)=-e^x\,dx)$, $y(0)=1$, then a value of $y(\log_e13)$ is:

Accepted Answer

$$\int\frac{dy}{2+y} = -\int\frac{e^x}{5+e^x}\,dx \implies \ln|2+y|+\ln|5+e^x|=\ln C.$$

Using $x=0,y=1$: $C=18$. At $x=\ln13$: $(2+y)(18)=18 \implies y=-1$.

Solutions · 1