Question

Solve the differential equation:

`x dy/dx = x·tan(y/x)+y`

Answer 1

`x dy/dx = x·tan(y/x)+y`  ...(I)

This is a homogeneous differential equation.

Put y = vx  ...(II)

Differentiate w. r. t. x, we get

∴ `dy/dx = v + x(dv)/dx` ...(III)

Put (II) and (III) in Eq. (I), it becomes,

`x(v + x(dv)/dx) = x tan((vx)/x) + vx`

divide by x, we get

∴ `v + x(dv)/dx = tan v + v`

∴ `x(dv)/dx = tan v`

∴ `(dv)/tan v = dx/x`

Integrating eq., we get

∴ `intcot v  dv = intdx/x`

∴ log (sin v) = log (x) + log c

∴ log (sin v) = log (x × c)

∴ sin v = cx

∴ `sin(y/x)` = cx is the solution.