How do I set this up to verify?

Question

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

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Answer 1 · 2018-02-26T18:13:46

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

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 · 2018-02-26T18:14:23

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!