Question

Find the particular solution of the differential equation $\frac{dy}{dx}+2y\tan x=\sin x$, given that $y=0$ when $x=\frac{\pi }{3}$.

Answer 1

Given the equation is $\frac{dy}{dx}+2ytanx=sinx$

Multiplying by $se{c}^{2}x$ on both sides

$\frac{dy}{dy}se{c}^2\left(x\right)+2yse{c}^2\left(x\right)tanx=sinxse{c}^2\left(x\right)$

This can be written as

$\frac{d\left(yse{c}^2\right(x\left)\right)}{dx}=tan\left(x\right)sec\left(x\right)$

Thus, integrating on both the sides, we get,

$\Rightarrow yse{c}^2\left(x\right)=sec\left(x\right)+c$ is the general solution

On substituting the values, $y=0$ when $x=\frac{\pi }{3}$ we get $c=2$

Thus the particular solution on substituting the values is $yse{c}^2\left(x\right)=sec\left(x\right)-2$