Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 an x \cdot \sec ^2 x$ such that $\mathrm{y}(0)=\frac{5}{4}$. Then $12\left(\mathrm{y}\left(\frac{\pi}{4} ight)-\mathrm{e}^{-2} ight)$ is equal to _______.

Question

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 	an x \cdot \sec ^2 x$ such that $\mathrm{y}(0)=\frac{5}{4}$. Then $12\left(\mathrm{y}\left(\frac{\pi}{4}ight)-\mathrm{e}^{-2}ight)$ is equal to _______.

Aicharya · Accepted Answer

$\begin{aligned} 	ext { I.F. } & =\mathrm{e}^{\int 2 \sec ^2 x d x} \ & =\mathrm{e}^{2 	an x}\end{aligned}$ Solution of diff. eq. $\begin{gathered}y \cdot e^{2 	an x}=\int e^{2 	an x}\left(2 \sec ^2 x+3 	an x \cdot \sec ^2 xight) d x \ y \cdot e^{2 	an x}=\int e^{2 	an x} \cdot\left(2 \sec ^2 xight) d x+\int e^{2 	an x} \cdot\left(3 	an x \cdot \sec ^2 xight) d x\end{gathered}$ y. $\mathrm{e}^{2 	an x}=\mathrm{e}^{2 	an x} \cdot 2 	an x-\int \mathrm{e}^{2 	an x} \cdot 2 \sec ^2 x 	imes 2 	an x d x+\int \mathrm{e}^{2 	an x} \cdot 3 	an x \cdot \sec ^2 x d x$ $\begin{aligned} & y \cdot e^{2 	an x}=2 	an x \cdot e^{2 	an x}-\int \mathrm{e}^{2 	an x} \cdot 	an x \sec ^2 x d x \ & y . e^{2 	an x}=2 	an x \cdot e^{2 	an x}-\frac{	an x \cdot e^{2 	an x}}{2}+\frac{e^{2 	an x}}{4}+C\end{aligned}$ $\begin{aligned} & y=2 	an x-\frac{	an x}{2}+\frac{1}{4}+\mathrm{Ce}^{-2 	an x} \ & x=0, y=\frac{5}{4} \ & c=1\end{aligned}$ $\mathrm{y}\left(\frac{\pi}{4}ight)=\frac{7}{4}+\mathrm{e}^{-2}$ Then $12\left(y\left(\frac{\pi}{4}ight)-\mathrm{e}^{-2}ight)=12\left(\frac{7}{4}ight)=21$

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan…

Solution