A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 4 years ago
Solve the differential equation \[ y'+3y2=4cos(\pi*x)+x^24e^{2x}\] using the method of undetermined coefficients
anonymous
 4 years ago
Solve the differential equation \[ y'+3y2=4cos(\pi*x)+x^24e^{2x}\] using the method of undetermined coefficients

This Question is Closed

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3\[y'+3y=4\cos(\pi x)+x^24e^{2x}+2\]now it's linear and should be doable with an integrating factor

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3\[\mu(x)=e^{\int 3dx}=e^{3x}\]multiply through...

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3\[(ye^{3x})'=4e^{3x}\cos(\pi x)+x^2e^{3x}+4e^{5x}+2e^{3x}\]\[\int (ye^{3x})'=ye^{3x}+C=\int4e^{3x}\cos(\pi x)+x^2e^{3x}+4e^{5x}+2e^{3x}dx\]some parts of this integral may get a little ugly, but nothing integration by parts can't handle

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3if you're not familiar with what I did on the left side you should acquaint yourself with solving linear equations with integrating factors

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0USING THE METHOD OF UNDETERMINED COEFFICIENTS. I know how to use the integrating factor method and I hate integration by parts.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Hatred is foolish. I distaste integrating by parts so please use the method of undetermined coefficients.

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3Undetermined coefficients for a firstorder DE? never heard of such a thing

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3well this was interesting, I didn't know you could use undetermined coefficients here, but here it is:

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3\[y'+3y=4\cos(\pi x)+x^24e^{2x}+2\]\[y_c=c_1e^{3}\]\[Y_p=A\cos(\pi x)+B\sin(\pi x)+Ce^{2x}+Dx^2+Ex+F\]\[Y'_p=A\pi\sin(\pi x)+B\pi\cos(\pi x)+2Ce^{2x}+2Dx+E\]plugging this into the equation we find that matching the coefficients on the left and right for\[Y_p'+3Y_p\]leads to the system\[\pi A+3B=0\]\[3A+B\pi=4\]\[5C=4\]\[3D=1\]\[3E+2D=0\]\[E+3F=2\]C, D, E, and F are found easily, and A and B can be found quickly using Cramer's rule. We find that\[A=\frac{12}{\pi^2+9}\]\[B=\frac{4\pi}{\pi^2+9}\]\[C=\frac54\]\[D=\frac13\]\[E=\frac23\]\[F=\frac{20}{27}\]so our solution is\[y(x)=c_1y_c+Y_p=\dots\]\[c_1e^{3x}+\frac{12}{\pi^2+9}\cos(\pi x)+\frac{4\pi}{\pi^2+9}\sin(\pi x)\frac45e^{2x}+\frac13x^2\frac29x+\frac{20}{27}\]actually that wasn't so bad. maybe even easier than integration by parts. Interesting!

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.3*the last number is\[+\frac{20}{27}\]in case you can't see it
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.