This question is inspired by this evaluation of an integral by Michael Penn. Suppose we have a differential equation $y''+p(x) y' +q(x)y =0$ with linearly independent solutions $y_1(x)$ and $y_2(x)$ over some interval $I$. Given that $p,q$ are continuous then $$ \mathrm{d}\left(\frac{y_2}{y_1} \right) = \frac{y_1 y_2' -y_1' y_2}{y_1^2} \mathrm{d}x= \frac{W(y_1, y_2)(x)}{y_1^2} \mathrm{d}x= W(y_1, y_2)(x_0) e^{-\int_{x_0}^{x}p(t)\, \mathrm{d}t}\frac{\mathrm{d}x}{y_1^2}, \quad x_0 \in I $$ where Abel's formula was used in the last step.
So if we now suppose that we have an integral like $\int_{a}^{b} \frac{\mathrm{d}x}{y_1^2(x)+y_2^2(x)}$ then we can do the following $$ \int_{a}^{b} \frac{\mathrm{d}x}{y_1^2(x)+y_2^2(x)} = \int_{a}^{b} \frac{1}{1+ \left( \frac{y_2}{y_1}\right)^2}\frac{\mathrm{d}x}{y_1^2 } \overset{u = y_2/y_1}{=}\frac{1}{W(y_1,y_2)(x_0)}\int_{\alpha}^{\beta}\frac{\mathrm{d}u}{\left(1+u^2\right)e^{-\int_{x_0}^{\left( \frac{y_2}{y_1}\right)^{-1}(u)}p(t)\, \mathrm{d}t}} $$ which is ugly in general, except for when $p =0$, and then the integral reduces to $$ \int_{a}^{b} \frac{\mathrm{d}x}{y_1^2(x)+y_2^2(x)} = \frac{\arctan\left(\frac{y_2(b)}{y_1(b)}\right)- \arctan\left(\frac{y_2(a)}{y_1(a)}\right)}{W(y_1,y_2)(x_0)} $$
One application of the above is for the integral $$ \int_{0}^{\infty} \frac{\mathrm{d}x}{x(I_n(x)^2 + K_{n}(x)^2)} $$ where $I,K$ are the modified Bessel functions. If we know that $x^2y'' -\left(x^2 + n^2 - \frac{1}{4}\right)y=0$ has solutions $\sqrt{x}I_n(x)$ and $\sqrt{x}K_{n}(x)$ then we can readily evaluate the integral as $$ \frac{\pi}{x\left(I_nK_{n-1} + I_nK_{n+1}+K_nI_{n-1} + K_nI_{n+1}\right)\Big\vert_{x_0}} = \frac{\pi}{2} $$
Does this technique have a name? And do you know some other examples of integrals that could be evaluated by a similar method? Thank you!
