Internal problem ID [12063]
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]]
\[ \boxed {t^{2} x^{\prime \prime }+t x^{\prime }-x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 1, x^{\prime }\left (1\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.0 (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: 0.119 (sec). Leaf size: 172
DSolve[{t^2*x''[t]+t*x[t]-x[t]==0,{x[1]==1,x'[1]==1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {\sqrt {t} \left (\left (\operatorname {BesselJ}\left (\sqrt {5},2\right )-\operatorname {BesselJ}\left (-1+\sqrt {5},2\right )+\operatorname {BesselJ}\left (1+\sqrt {5},2\right )\right ) \operatorname {BesselJ}\left (-\sqrt {5},2 \sqrt {t}\right )-\left (\operatorname {BesselJ}\left (-\sqrt {5},2\right )-\operatorname {BesselJ}\left (-1-\sqrt {5},2\right )+\operatorname {BesselJ}\left (1-\sqrt {5},2\right )\right ) \operatorname {BesselJ}\left (\sqrt {5},2 \sqrt {t}\right )\right )}{\operatorname {BesselJ}\left (\sqrt {5},2\right ) \left (\operatorname {BesselJ}\left (-1-\sqrt {5},2\right )-\operatorname {BesselJ}\left (1-\sqrt {5},2\right )\right )+\operatorname {BesselJ}\left (-\sqrt {5},2\right ) \left (\operatorname {BesselJ}\left (1+\sqrt {5},2\right )-\operatorname {BesselJ}\left (-1+\sqrt {5},2\right )\right )} \]