Finding general solution of non-linear, homogeneous ODE

[ohspyro89]

Homework Statement

Find the general solution to x³y'''-9x²y''+76xy'=0

Homework Equations

I'm kind of confused on where to start. I'd suppose you'd throw in y=e^mx but I'm not positive since there are Xs in there. Once I know what to do with the x terms, I can just find all the roots and that'll give me Y_c

Using y=y_c+y_p is the right way... right?

The Attempt at a Solution

I haven't got an attempt yet, because I'm unsure how to start!

 

[n1person]

You might want to try a series solution, meaning set $$y = \sum a_n x^n$$ and find the $$a_n$$

 

[ohspyro89]

I have no idea how to do that.

What we have covered is Undetermined Coefficients, variation of parameters, and Cauchy-Euler Equations...

We definitely haven't hit series stuff. Any other ideas?

 

[vela]

If those are the only methods you know so far, you might want to check with your instructor if there's a typo in the problem statement.

 

[ohspyro89]

Woops.

Should be

x³y'''-9x²y''+76xy'=0

 

[vela]

OK, so that's a Cauchy-Euler type of equation.

 

[HallsofIvy]

That is a "Cauchy-Euler" equation. Rather than using $e^{rx}$ you should try $x^r$ as a trial solution. This is because the substitution t= ln(x) will change this to a "constant coefficients" equation for which $e^{rt}$ is a good trial solution.

If $y= x^r$ then $y'= rx^{r-1}$, $y''= r(r-1)x^{r-2}$, and $y'''= r(r-1)(r-2)x^{r-3}$ so your equation comes to $x^3(r)(r-1)(r-2)x^{r-3}- 9x^2(r)(r-1)x^{r-2}+ 76x(r)x^{r-1}= (r(r-1)(r-2)- 9r(r-1)+ 76r)x^r= 0$. For all x except 0, you must have $r(r-1)(r-2)- 9r(r-1)+ 76r= 0$. Of course, it is easy to see that r= 0 is a solution to that so it quickly reduces to a quaratic.

 

[ohspyro89]

Thanks! I'm sure I'll be back with more questions on this problem.

But for another one, am I correct when doing undetermined coefficients that for 50e^6x-14cos(x)-175sin(x) that M=6, +-i, +-i?

 

[Ray Vickson]

Woops.

Should be

x³y'''-9x²y''+76xy'=0

It might be simpler to let y' = z and divide both sides by x (for x =/= 0), to get:
x^2 z'' - 9x z' + 76 z = 0.

RGV

 

[ohspyro89]

Am I on the right track?

Also, do I just derive the final function since I substituted y'=z?

 

[vela]

You made a mistake applying the quadratic equation.

To find y, you integrate the function you get for z(x). You may find you would have been better off finding y(x) directly.

 

[ohspyro89]

Well, I took the advice as it looked easier, but I see what you mean. Integrating this is stupid, thank god for TI-89.

I'm getting a terribly ugly answer though, but I think it's correct.

 

[ohspyro89]

How's this look? I fixed the quadratic I think.

 

[vela]

You dropped the arbitrary constants when you integrated. You still need them in your final answer, and you can absorb those numerical constants on each term into them. Other than that, it looks good.

You should try substituting the solution back into the original DE to make sure it works.

 

[ohspyro89]

Oh, I forgot to put them back in once I got the integral off of my calculator.

Will do though, thanks for the help! Differential equations will be the death of me, I'm tired of getting terrible grades.

 

[TylerH]

Wolfram Alpha is also good for checking answers.