Question

Solve the following differential equation:

`x ("d"y)/("d"x) = y - xcos^2(y/x)`

Answer 1

Given `x ("d"y)/("d"x) = y - x cos^2 y/x`

The equation can be written as

`("d"y)/("d"x) = (y - cos^2 y/x)/x` ..........(1)

This is a homogeneous differential equation.

y = vx

`("d"y)/("d"x) = "v"(1) + x "dv"/("d"x)`

Substituting `("d"y)/("d"x)` value in equation (1), we get

`"v" + (x"dv")/("d"x) = ("v"x - x cos^2 ((vx)/x))/x`

`"v" + (x"dv")/("d"x) = ("v"x - x cos^2("v"))/x`

`"v" + (x"dv")/("d"x) = x (("v" - cos^2"v"))/x`

`x "dv"/("d"x) = "v" - cos^2"v" - "v"`

`"dv"/("d"x) = (- cos^2"v")/x`

`"dv"/(cos^2"v") = (-"d"x)/x`

Integrating on both sides, we get

`int sec^2"v"  "d"x = - int ("d"x)/x`

tan v = – log x + log C

tan v = log C – log x

tan v = `log ("C"/x)`

e^{tan v} = `"C"/x`

C = xe^{tan v}  

C = `xe"^(tan  y/x)` is a required equation.