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!