Question

Solve the following differential equation:

`(1 + x^2) dy/dx + 2xy = 4x^2`

Answer 1

`(1 + x^2) dy/dx + 2xy = 4x^2`

`dy/dx + (2x)/((1 + x^2))y = (4x^2)/((1 + x^2))`

Comparing with `dy/dx + Py = Q`:

`P = (2x)/((1 + x^2)), Q = (4x^2)/((1 + x^2))`

Integrating factor:

I.F. = `e^(int Pdx)`

= `e^(int (2x)/(1 + x^2) dx`

= `e^(ln (1 + x^2)`

= (1 + x²)

Thus, the solution of the differential equation is,

`y(1 + x^2) = int (1 + x^2) Q dx + C`

`y(1 + x^2) = int (1 + x^2) (4x^2)/((1 + x^2)) dx + C`

`y(1 + x^2) = int 4x^2 dx + C`

`y(1 + x^2) = (4x^3)/3 + C`

`y = ((4x^3)/3 + C)/((1 + x^2))`