${\mathit{e}}^{\mathit{x}}$ tan y dx + $\left(2-{\mathit{e}}^{\mathit{x}}\right)\mathit{s}\mathit{e}{\mathit{c}}^2$ y dy
∫ $\frac{{\mathit{e}}^{\mathit{x}}}{\left({\mathit{e}}^{\mathit{x}}-2\right)}$ dx = ∫ $\frac{\mathit{s}\mathit{e}{\mathit{c}}^2\mathit{y}}{\mathit{t}\mathit{a}\mathit{n}\mathit{y}}$ dy
now, ∫ $\frac{\mathit{f}\prime \left(\mathit{x}\right)}{\mathit{f}\left(\mathit{x}\right)}$ dx = log |f (x)| + c
log $\left|\left({\mathit{e}}^{\mathit{x}}-2\right)\right|$ - log |tan y| + log |c| = 0
log $\left|\frac{\mathit{c}\left({\mathit{e}}^{\mathit{x}}-2\right)}{\mathit{t}\mathit{a}\mathit{n}\mathit{y}}\right|$ = 0
$\frac{\mathit{c}\left({\mathit{e}}^{\mathit{x}}-2\right)}{\mathit{t}\mathit{a}\mathit{n}\mathit{y}}$ = ${\mathit{e}}^0$ = 1
tan y = (c (${\mathit{e}}^{\mathit{x}}$ - 2))
now, x = 0 , y = $\frac{\mathit{\pi }}{4}$
1 = (x (1 - 2))
c = - 1
so
tan y = (- 1 (${\mathit{e}}^{\mathit{x}}$ - 2))
y = $\mathit{t}\mathit{a}{\mathit{n}}^{-1}$ (2 - ${\mathit{e}}^{\mathit{x}}$)
OR
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + 2y tan x = sin x
comparing with the standard form $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + Py = Q
P = 2 tan x , Q = sin x
Now,
I.F. = ${\mathit{e}}^{\int \mathit{P}\mathit{d}\mathit{x}}$ = ${\mathit{e}}^{\int 2\mathit{t}\mathit{a}\mathit{n}\mathit{x}\mathit{d}\mathit{x}}$ = ${\mathit{e}}^{2\log |\mathit{s}\mathit{e}\mathit{c}\mathit{x}|}$ = $\mathit{s}\mathit{e}{\mathit{c}}^2$ x
y $\mathit{s}\mathit{e}{\mathit{c}}^2$ x = sec x + c
0 = 2 + c ⇒ c = - 2 (Since when x = $\frac{\mathit{\pi }}{3}$ , y = 0)
y $\mathit{s}\mathit{e}{\mathit{c}}^2$ x = sec x - 2