Let y=y(x) be the solution of the differential equation (d y)/(d...

$$\begin{aligned} \text{ I.F. } & =\mathrm{e}^{\int 2 \sec ^2 x d x} \\ & =\mathrm{e}^{2 \tan x}\end{aligned}$$$<br/>Solution of diff. eq.<br/>$\begin{gathered}y \cdot e^{2 \tan x}=\int e^{2 \tan x}\left(2 \sec ^2 x+3 \tan x \cdot \sec ^2 x\right) d x \\ y \cdot e^{2 \tan x}=\int e^{2 \tan x} \cdot\left(2 \sec ^2 x\right) d x+\int e^{2 \tan x} \cdot\left(3 \tan x \cdot \sec ^2 x\right) d x\end{gathered}$<br/>y.$\mathrm{e}^{2 \tan x}=\mathrm{e}^{2 \tan x} \cdot 2 \tan x-\int \mathrm{e}^{2 \tan x} \cdot 2 \sec ^2 x \times 2 \tan x d x+\int \mathrm{e}^{2 \tan x} \cdot 3 \tan x \cdot \sec ^2 x d x$<br/>$$\begin{aligned} & y \cdot e^{2 \tan x}=2 \tan x \cdot e^{2 \tan x}-\int \mathrm{e}^{2 \tan x} \cdot \tan x \sec ^2 x d x \\ & y . e^{2 \tan x}=2 \tan x \cdot e^{2 \tan x}-\frac{\tan x \cdot e^{2 \tan x}}{2}+\frac{e^{2 \tan x}}{4}+C\end{aligned}$$<br/>$$\begin{aligned} & y=2 \tan x-\frac{\tan x}{2}+\frac{1}{4}+\mathrm{Ce}^{-2 \tan x} \\ & x=0, y=\frac{5}{4} \\ & c=1\end{aligned}$$<br/>$\mathrm{y}\left(\frac{\pi}{4}\right)=\frac{7}{4}+\mathrm{e}^{-2}$<br/>Then$12\left(y\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)=12\left(\frac{7}{4}\right)=21