 Solution

Two functions are linearly dependent if constants $c_1,c_2$, not both zero, satisfy $c_1{y}_1+c_2{y}_2=0$ identically. Differentiating also gives $c_1{\dot{y}}_1+c_2{\dot{y}}_2=0$. Since the determinant of these two equations is the nonzero Wronskian $W$, both constants must vanish. Hence the two solutions are linearly independent.

Differentiating the Wronskian and substituting the homogeneous linear differential equation givesIntegrating proves Abel's identity:For variation of parameters, impose ${\dot{a}}_1{y}_1+{\dot{a}}_2{y}_2=0$ on the proposed particular solution. Then $\dot{y}=a_1{\dot{y}}_1+a_2{\dot{y}}_2$, and substitution into the inhomogeneous differential equation leaves ${\dot{a}}_1{\dot{y}}_1+{\dot{a}}_2{\dot{y}}_2=f$. Solving this two-by-two linear system givesArbitrary constants of integration merely add a homogeneous solution.

For the constant positive coefficient $q={\omega }^2$, take $\omega >0$. Direct differentiation verifies that $y_1=\cos \omega t$ and $y_2=\sin \omega t$ solve the homogeneous linear differential equation; their Wronskian is $W=\omega$. The requested integrands areTaking $t_0=0$ yieldsThus $|a_1(t)|\sim t/(2\omega )$ grows with power one, whereas $a_2$ is bounded. Their combination simplifies to $y_p=-t\cos (\omega t)/(2\omega )+\sin (\omega t)/(2{\omega }^2)$; the final sine term is itself a homogeneous solution. The secular term is the response to resonance in a differential equation.