Let y=y(x) be the solution of the differential equation xdy+2(1-x) x+x(2-x)dx=0 such that …

Question

Let $y=y(x)$ be the solution of the differential equation $\sec xdy+{2(1-x)	an x+x(2-x)}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to:

Accepted Answer

$\frac{dy}{dx}=2(x-1)\sin x+({x}^{2}-2x)\cos x$

Now both side integrate

$\begin{gathered} y(x)=\int 2(x-1)\sin xdx+[({x}^{2}-2x)(\sin x)-\int (2x-2)\sin xdx] \ y(x)=({x}^{2}-2x)\sin x+\lambda \ y(0)=0+\lambda \Rightarrow 2=\lambda \ y(x)=({x}^{2}-2x)\sin x+2 \ y(2)=2 \end{gathered}$

Solutions · 1