Question

Verify the following:
Solution: y=x^2
differential eq: xy'-2y=x^3•e^x

Answer 1

x 2x -2 x^2= x^2* e^x
2x^2-2x^2=x^2 e^x
0=x^2 e^x no!
I am not sure

Answer 2

y'=2x
plug in:
\[x(2x)-2(x^2)=x^3e^x\]
\[2x^2-2x^2=x^3e^x\]
yep it is not

Answer 3

unless x=0

Answer 4

y=x^2
y'=2x
now,
xy'-2y = x (2x)-2(x^2) = 2x^2-2x^2=0

Answer 5

Yeah, so it's not verifiable.

Answer 6

yes , it is not verified

Answer 7

It's a linear differential equation:
\[xy'-2y=x^3e^x\]Arrange it into a solvable form:
\[y'-(2/x)y=x^2e^x\]Find the integrating factor:
\[\mu(x)=e^{\int\limits-(2/x)dx}=e^{-2\ln|x|}=e^{\ln|x|^{-2}}=x^{-2}\]Multiply everything by this factor:
\[(x^{-2}y)'=e^x\]Integrate both sides and solve for y:
\[y=x^{2}e^{x}+c\]You can see that the solutions don't match.