Question

Solve the following differential equation:

`xdy/dx=y-xsin^2(y/x)`, given that y(1) = `pi/6`

Answer 1

`xdy/dx=y-xsin^2(y/x)`

`dy/dx=y/x-sin^2(y/x)`

Which is a homogeneous diff. equation

y = vx

⇒ `dy/dx=v+x(dv)/dx`

⇒ `v+x(dv)/dx=(vx)/x-sin^2((vx)/x)`

⇒ `v+x(dv)/dx=v-sin^2v`

⇒ `(dv)/sin^2v=-dx/x`

integrating both sides,

⇒ `intcosec^2vdv=-intdx/x`

⇒ –cotv = –log|x| + C 

⇒ cot v = log|x| – C

Put v = `y/x`

cot `(y/x)` = log|x| – C

y(1) = `pi/6`; x = 1, y = `pi/6`

Put, cot `(pi)/6=log|1| -C`   ...[∵ log|1| = 0]

⇒ `sqrt3=-C`

⇒ C = `-sqrt3`

`cot(y/x)=log|x|+sqrt3`