Internal problem ID [10482]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page
125
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {t^{2} x^{\prime \prime }+x^{\prime } t +\left (t^{2}-\frac {1}{4}\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \frac {\cos \left (t \right )}{\sqrt {t}} \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 19
dsolve([t^2*diff(x(t),t$2)+t*diff(x(t),t)+(t^2-1/4)*x(t)=0,cos(t)/sqrt(t)],x(t), singsol=all)
\[ x \left (t \right ) = \frac {c_{1} \sin \left (t \right )}{\sqrt {t}}+\frac {c_{2} \cos \left (t \right )}{\sqrt {t}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 39
DSolve[t^2*x''[t]+t*x'[t]+(t^2-1/4)*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {e^{-i t} \left (2 c_1-i c_2 e^{2 i t}\right )}{2 \sqrt {t}} \\ \end{align*}