Question

Solve the differential equation

`x + y dy/dx` = x² + y²

Answer 1

`x + y dy/dx` = x² + y²     ...(i)

Put x² + y² = v

∴ `2x + 2y dy/dx = (dv)/dx`

∴ `2(x + y dy/dx) = (dv)/dx`

∴ `x + y dy/dx = 1/2 (dv)/dx`

∴ Given equation (i) becomes

`1/2 (dv)/dx` = v

∴ `(dv)/v` = 2 dx

∴ `int (dv)/v = int 2  dx`

∴ log v = 2x + c 

∴ log(x² + y²) = x + c

is the required solution of given D.E.