Question

Find the particular solution of the differential equation `(dy)/(dx) - 2xy = 3x^2 e^(x^2); y(0) = 5`. 

Answer 1

Given differential equation is `(dy)/(dx) - 2xy = 3x^2 e^(x^2)`

On comparing the above equation with `(dy)/(dx) + Py = Q,`

We get `P = -2x, Q = 3x^2e^(x^2)`

∴ I = `e^(intPdx) = e^(-int 2xdx)`

= `e^(-2(x^2/2))`

= `e^(-x^2)`

∴ `y.e^(-x^2) = int 3x^2e^(x^2).(e^(-x^2))dx + C`

or, `y.e^(-x^2) = 3int x^2dx + C`

or, `y/(e^(x^2)) = 3[x^3/3] + C`

or, `y/(e^(x^2)) = x^3 + C`

or, `y = e^(x^2)x^3 + Ce^(x^2)`

Given, y(0) = 5

∴ 5 = 0 + C `e^0` ⇒ C = 5

Thus, required solution is 

or `y = e^(x^2)(x^3 + 5)`