Internal problem ID [10715]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 19, CauchyEuler equations. Exercises page 174
Problem number: 19.1 (iv).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {t^{2} x^{\prime \prime }+x^{\prime } t -x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 1, x^{\prime }\left (1\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 5
dsolve([t^2*diff(x(t),t$2)+t*diff(x(t),t)-x(t)=0,x(1) = 1, D(x)(1) = 1],x(t), singsol=all)
\[ x \left (t \right ) = t \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 151
DSolve[{t^2*x''[t]+t*x[t]-x[t]==0,{x[1]==1,x'[1]==1}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\sqrt {t} \left (\left (\left (1+\sqrt {5}\right ) \operatorname {BesselJ}\left (\sqrt {5},2\right )-2 \, _0\tilde {F}_1\left (;\sqrt {5};-1\right )\right ) \operatorname {BesselJ}\left (-\sqrt {5},2 \sqrt {t}\right )+\left (2 \, _0\tilde {F}_1\left (;-\sqrt {5};-1\right )+\left (\sqrt {5}-1\right ) \operatorname {BesselJ}\left (-\sqrt {5},2\right )\right ) \operatorname {BesselJ}\left (\sqrt {5},2 \sqrt {t}\right )\right )}{2 \, _0\tilde {F}_1\left (;-\sqrt {5};-1\right ) \operatorname {BesselJ}\left (\sqrt {5},2\right )+\operatorname {BesselJ}\left (-\sqrt {5},2\right ) \left (2 \sqrt {5} \operatorname {BesselJ}\left (\sqrt {5},2\right )-2 \, _0\tilde {F}_1\left (;\sqrt {5};-1\right )\right )} \\ \end{align*}