Let y=y(x) be the solution of the differential equation d y~d x=2 x(x+y)^3-x(x+y)-1, y(0)=…

Question

Let $y=y(x)$ be the solution of the differential equation

$\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$.

Then, $\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals :

Accepted Answer

$\begin{aligned} & \frac{d y}{d x}=2 x(x+y)^3-x(x+y)-1 \\ & 	ext { Put } x+y=t \\ & \Rightarrow \frac{d y}{d x}=\frac{d t}{d x}-1 \\ & \frac{d t}{d x}-1=2 x(t)^3-x t\end{aligned}$

$\begin{aligned} \Rightarrow & \frac{d t}{2 t^3-t}=x d x \\ & \int \frac{1}{2 t^3-t} d t=\int x d x \\ \Rightarrow & \int \frac{t}{2 t^4-t^2} d t=\int x d x\end{aligned}$

$\begin{aligned} & t^2=z \\ & 2 t d t=d z \\ & \frac{1}{2} \int \frac{d z}{2 z^2-z}=\int x d x \\ & \ln \left|\frac{z-\frac{1}{2}}{z}ight|=x^2+c\end{aligned}$

$\ln \left|\frac{(x+y)^2-\frac{1}{2}}{(x+y)^2}ight|=x^2+c$

$y(0)=1 \Rightarrow c=\ln \left(\frac{1}{2}ight)$

$\Rightarrow \frac{(x+y)^2-\frac{1}{2}}{(x+y)^2}=e^{x^2} 	imes \frac{1}{2}$

$\frac{(x+y)^2-\frac{1}{2}}{(x+y)^2}=\sqrt{e} 	imes \frac{1}{2}$

$\Rightarrow(x+y)^2=\frac{1}{2-\sqrt{e}}$

Solutions · 1