I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation:
$$\frac{\mathrm{d}w_1}{\mathrm{d}t}w_1 + \frac{\mathrm{d}w_2}{\mathrm{d}t}w_2 + \frac{c}{I}(w_1^2 + w_2^2) = 0$$
Now I need to find $w_1^2 + w_2^2$ in terms of time that is how they change with respect to time in order to derive the relation that I want.
Since $$ \frac{dw_1}{dt}w_1+\frac{dw_2}{dt}w_2=\frac12\frac{d}{dt}(w_1^2+w_2^2), $$ your differential equation is equivalent to: $$\tag{1} \frac{du}{dt}=\alpha u, $$ with $$ u=w_1^2+w_2^2,\quad \alpha=-2\frac{c}{I}. $$ Solving (1) we get: $$ w_1^2(t)+w_2^2(t)=\beta\exp\left(-2\frac{ct}{I}\right), $$ where $$ \beta=w_1^2(0)+w_2^2(0). $$