Question

Solution of the differential equation `(x + y dy/dx)(x^2 + y^2)` = 1, is ______.

Options

• (x² + y²) = x + c

• (x² + y²)² = 2x + c

• (x² + y²)² = 4x + c

• (x² – y²)² = 4x + c

Answer 1

Solution of the differential equation `(x + y dy/dx)(x^2 + y^2)` = 1, is (x² + y²)² = 4x + c.

Explanation:

`(x + y dy/dx)(x^2 + y^2)` = 1

Put x² + y² = v

∴ `2x + 2y dy/dx = (dv)/dx`

∴ `x + y dy/dx = 1/2 * (dv)/dx`

∴ Given equation becomes,

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

∴  v dv = 2 dx

∴ `v^2/2` = 2x + c’

∴ v² = 4x + c where, c = 2c’

∴ (x² + y²)² = 4x + c