Question

Solve the differential equation.

Answer 1

\[x \frac{ dy }{ dx }-4y=4x^4e^x\]

Answer 2

\[\frac{ dy }{ dx }-\frac{ 4y }{ x }=4x^3e^x = > \frac{ dy }{ dx }+(\frac{ -4 }{ x })*y=4x^3e^x\]Integrating factor = \[e^{\int\limits_{}^{}(-4/x)dx}=e^{-4lnx}=e^{lnx^{-4}}=x^{-4}\]
Now, what to do for next step?

Answer 3

Multiply through by integrating factor, ya? :)
\(\large\rm x^{-4}y'-4x^{-5}y=\frac{4}{x}e^x\)

Answer 4

Ooo that causes a problem on the right side though :ooo
hmmm

Answer 5

Hmm, I think I got it now, I was just wondering how to implement the factor x^-4 into the equation

Answer 6

Oh but it appears to be the right approach.
Wolfram is indicating that the "exponential integral" Ei(x) is part of the final solution.

Are you sure you pasted the question correctly?
Seems like a weird problem to be assigned.

Answer 7

The end product, involves using an integral that you don't have to solve

Answer 8

\[x^{-4}*dy/dx + d(x^{-4})/dx*y = (4/x)e^x\]
\[\frac{ d(x^{-4}*y) }{ dx }= \frac{ 4 }{ x }e^x = > \int\limits_{}^{}d(y*x^{-4})=\int\limits_{}^{}\frac{ 4 }{ x }e^xdx\]\[y * (x^{-4}) = \int\limits_{}^{}\frac{ 4 }{ x }e^xdx+C\]

Answer 9

Right now, I'm seeing how to get rid of x^{-4}

Answer 10

\[\large\rm \int\limits \frac{e^x}{x}dx=Ei(x)\]So I guess what you have right now is,\[\large\rm y x^{-4}=4Ei(x)+C\]

Answer 11

Okay, got it, just had to multiply x^{-4} to both sides,

\[y = (\int\limits_{}^{}\frac{ 4e^x }{ x }dx+C)x^{-4}\] which is the correct answer.

Answer 12

No no, multiply by x^4.

Answer 13

Okh yea, you multiply by x^4 to get rid of the x^{-4} on the left side,

left side -> y * (x^(-4+4)), so the finalized answer is \[y = (\int\limits_{}^{}\frac{ 4e^x }{ x }dx+C)x^4\]

Answer 14

Thanks for the help! :D

Answer 15

Would dividing both sides by x^{-4} also work?

Answer 16

Yes yes that works well :)
\(\large\rm \frac{1}{x^{-4}}=x^4\)

Answer 17

I see, I see, much appreciated for your wisdom and expertise. :)