Internal problem ID [1763]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.4, The method of variation of parameters. Page 154
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1}-t^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(diff(y(t),t$2)-2*t/(1+t^2)*diff(y(t),t)+2/(1+t^2)*y(t)=1+t^2,y(t), singsol=all)
\[ y \relax (t ) = c_{2} t +\left (t^{2}-1\right ) c_{1}+\frac {1}{2}+\frac {t^{4}}{6} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 33
DSolve[y''[t]-2*t/(1+t^2)*y'[t]+2/(1+t^2)*y[t]==1+t^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{6} \left (t^2+3\right ) t^2+c_2 t-c_1 (t-i)^2 \\ \end{align*}