63.13.5 problem 6

Internal problem ID [13176]
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
Date solved : Tuesday, January 28, 2025 at 05:12:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} x^{\prime \prime }+t x^{\prime }+\left (t^{2}-\frac {1}{4}\right ) x&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} x&=\frac {\cos \left (t \right )}{\sqrt {t}} \end{align*}

✓ Solution by Maple

Time used: 0.160 (sec). Leaf size: 17

dsolve([t^2*diff(x(t),t$2)+t*diff(x(t),t)+(t^2-1/4)*x(t)=0,cos(t)/sqrt(t)],singsol=all)
 

\[ x \left (t \right ) = \frac {c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )}{\sqrt {t}} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 39

DSolve[t^2*D[x[t],{t,2}]+t*D[x[t],t]+(t^2-1/4)*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {e^{-i t} \left (2 c_1-i c_2 e^{2 i t}\right )}{2 \sqrt {t}} \]