Diff equation (xy' - 2y = x^3 e^x)

tapout1829

New member
Hey guys i need help to do this problem:

Determine wheather the function is a solution of the differential equation --> xy' - 2y = x^3e^x

y = x^2e^x is the question

thanks guys :roll:

galactus

Super Moderator
Staff member
To see if $$\displaystyle y=x^{2}e^{x}$$ is a solution, take it and its derivative

and sub into the left side of your DE. If you get the right side, then it's a

solution.

If you need to know how to attempt deriving the answer, then first find your integrating factor.

$$\displaystyle \L\\x\frac{dy}{dx}-2y=x^{3}e^{x}$$

Divide through by x:

$$\displaystyle \L\\\frac{dy}{dx}-\frac{2}{x}y=x^{2}e^{x}$$

Now, can you find the integrating factor and finish?. It ain't too bad.

soroban

Elite Member
Re: Diff equation

Hello, tapout1829!

Are you in self-study?
This should have been explained in your class.

Determine whether $$\displaystyle y\:=\:x^2e^x$$ is a solution of the differential equation: $$\displaystyle \,xy'\,-\,2y\:=\:x^3e^x$$
This problem is similar to one you had in Algebra I.
$$\displaystyle \;\;$$Determine whether $$\displaystyle x\,=\,2$$ is a solution of: $$\displaystyle \,2x^2\,-\,x\:=\:6$$

We "plug in" $$\displaystyle x\,=\,2$$ and see if we get a true statement, right?
We replace every $$\displaystyle x$$ with $$\displaystyle 2$$ and see what we get.

The left side is: $$\displaystyle \,2\cdot2^2\,-\,2$$ . . . Does this equal the right side?
We have: $$\displaystyle \,8\,-\,2\:=\:6$$ . . . yes!

They already did the hard work (solving for $$\displaystyle x$$); we just had to verify the answer.

This is problem is the same.
They already solved the differential equation,
$$\displaystyle \;\;$$and they want us to verify their answer.

We have: $$\displaystyle \,xy'\,-\2y\:=\:x^3e^x$$
Answer: $$\displaystyle y\:=\:x^2e^x$$

"Plug in" the answer and see if we get a true statement.
Replace $$\displaystyle y$$ with $$\displaystyle x^2e^x$$ . . . Replace $$\displaystyle y'$$ with $$\displaystyle x^2e^x\,+\,2xe^x$$

The left side is: $$\displaystyle \,x\left(x^2e^x\,+\,2xe^x)\,-\,2(x^2e^x)$$ . . . Does this equal the right side?

We have: $$\displaystyle \,x^3e^x\,+\,2x^2e^x \,-\,2x^2e^x\;=\;x^3e^x$$ . . . yes!