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

Question

Let $y=y(x)$ be the solution of the differential equation

$\frac{dy}{dx}+3({	an }^{2}x)y+3y={\sec }^{2}x,y(0)=\frac{1}{3}+{e}^{3}$. Then $y(\frac{\pi }{4})$ is equal to :

Accepted Answer

$\begin{gathered} \frac{dy}{dx}+3({	an }^{2}x)y+3y={\sec }^{2}x \ \Rightarrow \frac{dy}{dx}+3{\sec }^{2}xy={\sec }^{2}x \ ={e}^{\int 3{\sec }^{2}xdx} \ \, ={e}^{3	an x} \end{gathered}$

$\begin{gathered} y\cdot {e}^{	an x}=\int {e}^{3	an x}\cdot {\sec }^{2}xdx+c \ y\cdot {e}^{3	an x}=\frac{{e}^{3	an x}}{3}+c \ f(0)=\frac{1}{3}+{e}^{3} \ \Rightarrow (\frac{1}{3}+{e}^{3})=\frac{1}{3}+c \ \Rightarrow c={e}^{3} \ 	herefore y\cdot {e}^{3	an x}=\frac{{e}^{3	an x}}{3}+{e}^{3} \ x=\frac{\pi }{4} \end{gathered}$

$y{e}^{3}=\frac{{e}^{3}}{3}+{e}^{3}\Rightarrow y=\frac{4}{3}$

Solutions · 1