Rewriting $y^{\prime \prime}(t)+\lambda^{2} y(t)+g(y(t))=0$ into integral form using variation-of-constant formula.

Question

Consider the oscillatory second order ordinary differential equation (ODE) $$ y^{\prime \prime}(t)+\lambda^{2} y(t)+g(y(t))=0, \quad 0<t \leq T, $$ with the initial data $$ y(0)=\alpha, \quad y^{\prime}(0)=\beta, $$ where $\lambda>>1, \alpha$ and $\beta$ are given constants, and $g(y)$ is a given Lipschitz continuous function. Re-write the above ODE into its equivalent integral formulation near $t=t_{n}$ with $t=t_{n}+s$ for $s \in \mathbb{R}$ via the variation-of-constant formula.$$$$ I think to use the variation-of-constant formula, we need to consider the equation $ y^{\prime \prime}(t)+\lambda^{2} y(t)=0$ first. This is solvable. However the term $g(y(t))$ involves $y(t)$, and I don't know how to proceed. I think I have some misunderstanding on variation-of-constant formula, and how to Re-write the above ODE into integral using it? Thanks!

Answer 1

In this case you can simplify the variation-of-constants or application of the Green kernel using that the characteristic roots of the (explicitly) linear part are imaginary. Set $$z=λy+iy',$$ then $$ z'=λy'+iy''= λy'-iλ^2y-ig(y)=-iλz-ig(y) $$ Combine the linear terms with an integrating factor $$ e^{iλt}z(t)-z(0)=-i\int_0^te^{iλs}g(y(s))\,ds. $$ Isolate $z(t)$ and take the real part to get $$ λy(t)=\underbrace{λy(0)\cos(λt)+y'(0)\sin(λt)}_{=Re(z(0)e^{-iλt})}-\int_0^t\sin(λ(t-s))g(y(s))\,ds. $$