Let the solution curve y=y(x) of dydx-3x^5^-1(x^3)(1+x^6)^3/2y = 2x(x^3-^-1x^3)(1+x^6) pas…

Question

Let the solution curve $y=y(x)$ of $\dfrac{dy}{dx}-\dfrac{3x^5	an^{-1}(x^3)}{(1+x^6)^{3/2}}y = \dfrac{2x\exp\left(x^3-	an^{-1}x^3ight)}{(1+x^6)}$ pass through the origin. Then $y(1)$ is:

Accepted Answer

I.F. $=e^{\tan^{-1}x^3-x^3/(1+x^6)\dots}$; the ODE is engineered so that
$$y\cdot e^{\left(\tan^{-1}x^3-x^3\right)/(1+x^6)} = x^2+c.$$

Using $y(0)=0 \Rightarrow c=0$. At $x=1$: $y(1)\cdot e^{(\pi/4-1)/\sqrt2}=1 \implies y(1)=e^{(4-\pi)/(4\sqrt2)}$.

Solutions · 1