Question

How do you use the differential equation `dy/dx=18x^2(2x^3+1)^2` to find the equation of the function given point (0,4)?

Answer 1

The function has equation

`y = (2x^3 + 1)^3 - 23`

Explanation:

In this problem, I figure that a substitution would be much simpler than integration by parts. Notice how the binomial within the parentheses has a highest term that is a degree higher than the `18x^2`? This condition will allow the substitution to be effective.

Let `u = 2x^3 + 1`. Then `du = 6x^2dx` and `dx = (du)/(6x^2)`.

`dy/dx = 18x^2(u^2) * (du)/(6x^2)`

`dy/dx = 3u^2 du`

To solve for the function `y`, we must integrate on both side of the equation.

`int (dy/dx) = int (3u^2)du`

Now use `int x^ndx = x^(n + 1)/(n + 1) + C`

`y = u^3 + C`

You can now reverse the substitution.

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

All that is left to do is to solve for `C`.

`4 = (2(1)^3 + 1)^3 + C`

`4 = (2 + 1)^3 + C`

`4 - 27 = C`

`C = -23`

Hopefully this helps!