Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y([])…

Question

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

equation xdy = (y + x3 cosx)dx with y($$\pi$$) = 0, then $$y\left( {{\pi  \over 2}} \right)$$ is equal to :

Accepted Answer

$$xdy = (y + {x^3}\cos x)dx$$

$$ \Rightarrow $$ $$xdy = ydx + {x^3}\cos xdx$$

$$ \Rightarrow $$ $${{xdy - ydx} \over {{x^2}}} = {{{x^3}coxdx} \over {{x^2}}}$$

$$ \Rightarrow $$ $${d \over {dx}}\left( {{y \over x}} \right) = \int {x\cos xdx} $$

$$ \Rightarrow {y \over x} = x\sin x - \int {1.\sin xdx} $$

$$ \Rightarrow $$ $${y \over x} = x\sin x + \cos x + C$$

$$ \Rightarrow 0 =  - 1 + C \Rightarrow C = 1,x = \pi ,y = 0$$

so, $${y \over x} = x\sin x + \cos x + 1$$

$$y = {x^2}\sin x + x\cos x + x$$

$$x = {\pi  \over 2}$$

$$y\left( {{\pi  \over 2}} \right) = {{{\pi ^2}} \over 4} + {\pi  \over 2}$$

Solutions · 1