Question

Find the general solution of the differential equation. (Use C for the constant of integration.) dy dx = 36 − 12x2 x3 − 9x + 3

Answer 1

`y = -4ln(x^3 - 9x + 3) + C\\`

Explanation:

Given the differential equation
`(dy)/(dx) = (36-12x^2)/(x^3-9x+3)\\ \\`, we will use the variable separable method to solve the differential equation as shown;

`(dy)/(dx) = (36-12x^2)/(x^3-9x+3)\\ \\dy = (36-12x^2)/(x^3-9x+3)dx\\ \\\\\\integrate \ both \ sides\\\\\int\limits dy = \int\limits(36-12x^2)/(x^3-9x+3)dx\\ \\\\using\ substitution\ method \ to \ solve \ RHS\\\\\int\limits(36-12x^2)/(x^3-9x+3)dx\\\\let \ u = x^3-9x+3; du/dx = 3x^2-9\\\\dx = du/3x^2-9\\\\\int\limits(36-12x^2)/(x^3-9x+3)dx\\\\ = \int\limits(36-12x^2)/(u)*(du)/(3x^2-9) \\\\= \int\limits(12(3-x^2))/(u)*(du)/(3(x^2-3))\\\\`

`= \int\limits(-12(x^2-3))/(u)*(du)/(3(x^2-3))\\\\= -4\int\limits (du)/(u)\\ = -4lnu + C\\= -4ln(x^3 - 9x + 3)`

The differential solution becomes:

`y = -4ln(x^3 - 9x + 3)`
`+C`