Let y=y(x) solve dydx=2x(x+y)^3-x(x+y)-1, y(0)=1.

Question

Let $y=y(x)$ solve $\dfrac{dy}{dx}=2x(x+y)^3-x(x+y)-1$, $y(0)=1$. Then $\left(\dfrac{1}{\sqrt2}+y\left(\dfrac{1}{\sqrt2}\right)\right)^2$ equals:

Accepted Answer

Substituting $u=x+y$ transforms the ODE into a separable equation in $u,x$; solving with $y(0)=1$ (i.e. $u(0)=1$) and evaluating at $x=1/\sqrt2$ gives the stated value.

Solutions · 1