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

2 Answers
Feb 26, 2018

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

Feb 26, 2018

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!