Internal problem ID [1751]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 17
dsolve(t^2*diff(y(t),t$2)+t*diff(y(t),t)+(t^2-1/4)*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{\sqrt {t}} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 39
DSolve[t^2*y''[t]+t*y'[t]+(t^2-1/4)*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {e^{-i t} \left (2 c_1-i c_2 e^{2 i t}\right )}{2 \sqrt {t}} \]