Homogeneous linear ODEs with Constant Coefficients

[oasi]

do you have a idea about it?can you help me

http://img17.imageshack.us/img17/1156/18176658.png

 

[Dustinsfl]

do you have a idea about it?can you help me

http://img17.imageshack.us/img17/1156/18176658.png

For example, take $ay' + by = 0$. Solving for y' yields
$$
y' = -\frac{b}{a}y = ky
$$
where k = -b/a.

The only nontrivial function whose derivative is a constant multiple of itself is an exponential function.

 

[Mathwebster]

But $y = \sin x$ could work. For example,

$y'' + y = 0$

has as one solution $y = \sin x$.

 

[Dustinsfl]

But $y = \sin x$ could work. For example,

$y'' + y = 0$

has as one solution $y = \sin x$.

As long as the boundaries aren't $y'(0) = 0$ and $y'\left(\frac{\pi}{2}\right) = 0$ then y = 0. :)

But $y = A\sin x + B\cos x = e^0\left(A\sin x + B\cos x\right)$ is also a solution of the non boundary value problem.