Math 6410 • ODE

\textbf{Dependence on Initial Conditions.}
Consider the scalar initial value problem
\[
\begin{aligned}
  y'' &= y-y^3,\\
  y(0) &= a,\\
  y'(0) &= b.
\end{aligned}
\]
Denote the dependence of \(y(t)\) on its initial conditions by \(y(t;a,b)\). Without
explicitly solving for \(y(t;a,b)\), find
\(\frac{\partial y}{\partial a}(t;0,0)\) and
\(\frac{\partial y}{\partial b}(t;0,0)\), then write down a first-order approximation for
\(y(t;a,b)\) which is valid for small \(\lvert a\rvert,\lvert b\rvert\).