Let y=y(x) be the solution curve of the differential equation y(dy)/(dx)+2x y=x^3 y,y(1)=0…

Question

Let $y=y(x)$ be the solution curve of the differential equation $\sec y\frac{dy}{dx}+2x\sin y={x}^{3}\cos y,y(1)=0$. Then $y(\sqrt{3})$ is equal to:

Accepted Answer

$\begin{gathered} \sec y\frac{dy}{dx}+2x\sin y={x}^{3}\cos y \ \Rightarrow {\sec }^{2}y\frac{dy}{dx}+2x	an y={x}^{3} \end{gathered}$

Let $z=	an y$

$\begin{gathered} \frac{dz}{dx}={\sec }^{2}y\frac{dy}{dx} \ \Rightarrow \frac{dz}{dx}+2xz={x}^{3} \end{gathered}$

$\begin{gathered} ={e}^{{x}^{2}} \ \Rightarrow z.{e}^{{x}^{2}}=\int {e}^{{x}^{2}}\cdot {x}^{3}dx+c \end{gathered}$

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

$\Rightarrow 	an (0)\cdot e=\frac{1}{2}(1\cdot e-e)+c$

$\Rightarrow c=0$

$\Rightarrow 	an y=\frac{{x}^{2}-1}{2}$

$f(x)={	an }^{-1}(\frac{{x}^{2}-1}{2})\Rightarrow f(\sqrt{3})=\frac{\pi }{4}$

Solutions · 1