Differential equations and proof

Question

I have solved this. But I don't know how to came up with the (1\2) beside m. I have tried I but I was unable to do that. Please help me in this.

Answer 1

The trick is to just to realize that $\int x \dot{x}dt = \frac{x^2}{2} + C$ where $C$ is an integration constant (try integrating by parts with $u(t) = x(t)$ and $v(t) = \dot{x} dt$ if you have troubles with it). The same is true if we change $x \mapsto \dot{x}$ and $\dot{x} \mapsto \ddot{x}$. So when you multiply by $\frac{dx}{dt}$ and the integrate, you are just creating two integrals of the same functional form: $$ \begin{aligned} &\int \dot{x} \ddot{x} dt + \omega^2\int x \dot{x} dt = 0 \\ &\iff \frac{1}{2} \left( \frac{dx}{dt} \right) ^2 + C_1 + \omega^2 \left( \frac{1}{2}x^2 + C_2 \right) = 0 \end{aligned} $$

You get to the result just by using the fact that $\omega^2 = \frac{k}{m}$, multiplying by $m$ and summing all the corresponding constants to $E$.

I hope it helps :)