Internal problem ID [11486]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {x^{\prime \prime }+t^{2} x^{\prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 1.297 (sec). Leaf size: 63
dsolve([diff(x(t),t$2)+t^2*diff(x(t),t)=0,x(0) = 0, D(x)(0) = 1],x(t), singsol=all)
\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {t^{3}}{3}} \sqrt {t}\, \left (4 \,3^{\frac {5}{6}} \left (t^{3}\right )^{\frac {1}{6}}+9 \,{\mathrm e}^{\frac {t^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right )\right ) 3^{\frac {1}{6}} \left (\left \{\begin {array}{cc} \frac {1}{1-i \sqrt {3}} & t <0 \\ \frac {1}{2} & 0\le t \end {array}\right .\right )}{6} \]
✓ Solution by Mathematica
Time used: 0.153 (sec). Leaf size: 43
DSolve[{x''[t]+t^2*x'[t]==0,{x[0]==0,x'[0]==1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {t^2 \operatorname {Gamma}\left (\frac {1}{3}\right )-\left (t^3\right )^{2/3} \Gamma \left (\frac {1}{3},\frac {t^3}{3}\right )}{3^{2/3} t^2} \]