Question

Find the solution of the differential equation 3e^(3x)dy/dx=â9x/y^2

Answer 1

We want to solve

`3e^(3x) (dy)/(dx) = (9x)/(y^(2))`

The ODE is separable into the form

`3y^(2) dy = 9xe^(-3x) \\\\ y^(2) dy = 3xe^(-3x)dx`

Integrate.

`(1)/(3) \int y^(2) dy = \int x e^(-3x) dx \\\\ (y^(3))/(9) = -(xe^(-3x))/(3) + (1)/(3) \int e^(-3x) dx \\\\ = - (x)/(3)e^(-3x) - (1)/(9) e^(-3x) + c`


`y^(3) = -Ce^(-3x) (1+3x) \\\\ y = - ce^(-x) \sqrt[3]{1+3x}`

Answer:
`y = -ce^(-x) \sqrt[3]{1+3x}, \,\,\, c=constant`