anonymous
  • anonymous
Solve the differential equation \[ y'+3y-2=4cos(\pi*x)+x^2-4e^{2x}\] using the method of undetermined coefficients
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
TuringTest
  • TuringTest
\[y'+3y=4\cos(\pi x)+x^2-4e^{2x}+2\]now it's linear and should be doable with an integrating factor
TuringTest
  • TuringTest
\[\mu(x)=e^{\int 3dx}=e^{3x}\]multiply through...
TuringTest
  • TuringTest
\[(ye^{3x})'=4e^{3x}\cos(\pi x)+x^2e^{3x}+4e^{5x}+2e^{3x}\]\[\int (ye^{3x})'=ye^{3x}+C=\int4e^{3x}\cos(\pi x)+x^2e^{3x}+4e^{5x}+2e^{3x}dx\]some parts of this integral may get a little ugly, but nothing integration by parts can't handle

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

TuringTest
  • TuringTest
if you're not familiar with what I did on the left side you should acquaint yourself with solving linear equations with integrating factors
anonymous
  • anonymous
USING THE METHOD OF UNDETERMINED COEFFICIENTS. I know how to use the integrating factor method and I hate integration by parts.
anonymous
  • anonymous
Hatred is foolish. I distaste integrating by parts so please use the method of undetermined coefficients.
TuringTest
  • TuringTest
Undetermined coefficients for a first-order DE? never heard of such a thing
TuringTest
  • TuringTest
well this was interesting, I didn't know you could use undetermined coefficients here, but here it is:
TuringTest
  • TuringTest
\[y'+3y=4\cos(\pi x)+x^2-4e^{2x}+2\]\[y_c=c_1e^{-3}\]\[Y_p=A\cos(\pi x)+B\sin(\pi x)+Ce^{2x}+Dx^2+Ex+F\]\[Y'_p=-A\pi\sin(\pi x)+B\pi\cos(\pi x)+2Ce^{2x}+2Dx+E\]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
  • TuringTest
*the last number is\[+\frac{20}{27}\]in case you can't see it

Looking for something else?

Not the answer you are looking for? Search for more explanations.