Question

How do I set this up to verify?

Verify that the given function is a solution of the given differential equation for A.
y=Ax+x^3, x(dy/dx)=y+2x^3

Answer 1

Take `dy/dx`, plug it into the differential equation, plug `y=Ax+x^3` into the right side, simplify both sides.

Explanation:

First, calculate `dy/dx:`

`dy/dx=A+3x^2` as per the Power Rule. We just treat `A` as some constant.

Now, plug in `dy/dx` into the differential equation `x(dy/dx)=y+2x^3`

`x(A+3x^2)=y+2x^3`

Recall that `y=Ax+x^3`. Plugging that into the right side yields:

`x(A+3x^2)=Ax+x^3+2x^3`

`x(A+3x^2)=Ax+3x^3`

Multiplying `x` through the left side yields:

`Ax+3x^3=Ax+3x^3`

Answer 2

We have:

`dy/dx = A + 3x^2`

Now substitute:

`x(3x^2 + A) = Ax + x^3 + 2x^3`

`3x^2 + Ax = Ax + 3x^3`

This is clearly true, therefore the `y = Ax + x^3` is indeed a solution to the given differential equation.

Hopefully this helps!