Internal problem ID [11488]
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 2.4.1 Cauchy-Euler equations.
Exercises page 120
Problem number: 1(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x^{\prime \prime }+\frac {x}{t^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
dsolve(diff(x(t),t$2)=-1/t^2*x(t),x(t), singsol=all)
\[ x \left (t \right ) = c_{1} \sqrt {t}\, \sin \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right ) \sqrt {t} \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 42
DSolve[x''[t]==-1/t^2*x[t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \sqrt {t} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \log (t)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \log (t)\right )\right ) \]