# Math Help - second order ODE

1. ## second order ODE

Hello,

I have this problem:

solve this ODE

$u'' + u = 0$

so the Characteristic Equation is going to be:
$r^2 + 1 = 0$

i'm going to get a complex root correct? I'm really unsure on how to approach complex roots.

2. Originally Posted by dlbsd
Hello,

I have this problem:

solve this ODE

$u'' + u = 0$

so the Characteristic Equation is going to be:
$r^2 + 1 = 0$

i'm going to get a complex root correct? I'm really unsure on how to approach complex roots.
If the roots of the auxillary equation are $r = a \pm i b$ then the homogenous solution to the differential equation is $u = e^{at} (A \cos (bt) + B \sin (bt))$.

3. Since, u''=-u

A function differentiated twice gives the negative of the same function, so it can be either A*cosine(bx) or B*sine(bx). By principle of superposition of the results (which is saying if two solutions are there bundle them up together)... u(x)=A cos (bx) + B sin (bx)