1
$\begingroup$

Find the general solution of the nonhomogeneous differential equation

$$y'' + 4y =\sin(\alpha t).$$

For all possible real $\alpha$ decide which values of $\alpha$ gives a limited amount of solutions.

I found the homogeneous solution which yields: $y_h= C_1\sin(2t) + C_2\cos(2t)$

Any tips/solutions?

$\endgroup$
3
  • $\begingroup$ What do you mean by limited amount of solutions? $\endgroup$
    – David H
    May 20 '14 at 12:15
  • $\begingroup$ doesnt this get infinite amount of solutions since I have alpha? or something? $\endgroup$
    – gger234
    May 20 '14 at 12:24
  • $\begingroup$ So now you need to find a particular solution of the inhomogeneous equation. This will be of the form $A\sin(at)+B\cos(at)$ where you work out the values of $A$ and $B$ by substituting into the equation, except that if $a=2$ then you have to use $t(A\sin(at)+B\cos(at))$. $\endgroup$ May 20 '14 at 12:50
1
$\begingroup$

Hint:

$y=sin(\alpha t)$ is a solution of $$\alpha^2y+y''=0$$ $$(D^2+\alpha^2)y=0$$ so apply

$$(D^2+\alpha^2)(y''+4y)=(D^2+\alpha^2)(sin(\alpha t))$$ $$(D^2+\alpha^2)(D^2+4)(y)=0$$ $$(D^4+(\alpha^2+4)D^2+4\alpha^2)y=0 $$

Thus, we have a homogenius equation with characterestic polynomial $$\lambda^4+(\alpha^2+4)\lambda^2+4\alpha^2=0 $$

Note: I used the $D$ as derivative operator.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.