If y = y(x) is the solution of the differential equation [dy dx] + (tan x) y = sin x, [0 x…

Question

If y = y(x) is the solution of the differential equation

$${{dy} \over {dx}}$$ + (tan x) y = sin x, $$0 \le x \le {\pi  \over 3}$$, with y(0) = 0, then $$y\left( {{\pi  \over 4}} \right)$$ equal to :

Accepted Answer

Integrating Factor $$= {e^{\int {\tan x\,dx} }} = {e^{\ln (\sec x)}} = \sec x$$

$$y\sec x = \int {(\sin x)\sec x\,dx = \ln (\sec x) + C} $$

$$y(0) = 0 \Rightarrow C = 0$$

$$ \therefore $$ $$y = \cos x\ln |\sec x|$$

$$y\left( {{\pi  \over 4}} \right) = {1 \over {\sqrt 2 }}\ln \left( {\sqrt 2 } \right) = {1 \over {2\sqrt 2 }}\ln 2$$

Solutions · 1