I have derived the following solution to a given differential equation, regarding simple harmonic motion: $$x(t)=c_1 \cos\left(\sqrt{\frac{k}{m}}t\right)+c_2\sin\left(\sqrt{\frac{k}{m}}t\right).$$ I am trying to find the constants $c_1$ and $c_2$.
I have found $c_1$, by considering: $$x(0)=c_1 \cos\left(\sqrt{\frac{k}{m}}(0)\right)+c_2\sin\left(\sqrt{\frac{k}{m}}(0)\right).$$ which leads to $$x_0=c_1.$$ However, I am unsure as how to find the second one: I have attempted to find it but I am convinced that I am wrong: Considering: $$x'(0)=v(0)=-c_1\sqrt{\frac{k}{m}}\sin \left(\sqrt{\frac{k}{m}}(0)\right)+c_2\cos \left(\sqrt{\frac{k}{m}}(0)\right)\sqrt{\frac{k}{m}}$$ This would return a constant, $c_2$ of: $$v_0\sqrt{\frac{m}{k}}.$$ As I said, I am convinced this is wrong, if anyone could offer me some advice as to how to find the actual value, that'd be great. Thanks.
Why you are convinced that your computations are wrong ? Everything is fine:
$c_1=x_0$ and $c_2= v_0\sqrt{\frac{m}{k}}.$