Question

Solve the differential equation `x (d^2 y)/(dx^2) +(x-1) (dy)/(dx)-y = x^2` ?

Answer 1

`y = C_0 e^-x + C_1(1-x)+x^2`

Explanation:

This is a linear differential equation so it can be solved as the addition of the homogeneous solution

`x (d^2 y_h)/(dx^2) +(x-1) (dy_h)/(dx)-y = 0` and a particular solution

`x (d^2 y_p)/(dx^2) +(x-1) (dy_p)/(dx)-y = x^2`

For the homogeneous solution, substituting `y_h = c_0e^(lambda x)` we obtain

`c_0e^(lambda x) (1 + lambda) (lambda x-1) = 0` and then

`lambda = -1`

For the particular solution we propose

`y_p = c_1+c_2x+c_3x^2`

and after substituting

`(c_3-1)x^2-c_1-c_2=0` then

`c_3 = 1, c_1 = -c_2`

or

`y_p = c_1-c_1x + x^2`

and the general solution

`y = y_h+y_p = C_0 e^-x + C_1(1-x)+x^2`