57.11.9 problem 2

Internal problem ID [14443]
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
Date solved : Thursday, October 02, 2025 at 09:37:17 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{\prime \prime }+t^{2} x^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.701 (sec). Leaf size: 62

ode:=diff(diff(x(t),t),t)+t^2*diff(x(t),t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\sqrt {t}\, 3^{{1}/{6}} {\mathrm e}^{-\frac {t^{3}}{6}} \left (4 \,{\mathrm e}^{-\frac {t^{3}}{6}} 3^{{5}/{6}} \left (t^{3}\right )^{{1}/{6}}+9 \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right )\right ) \left (\left \{\begin {array}{cc} \frac {1}{1-i \sqrt {3}} & t <0 \\ \frac {1}{2} & 0\le t \end {array}\right .\right )}{6} \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 43

ode=D[x[t],{t,2}]+t^2*D[x[t],t]==0; 
ic={x[0]==0,Derivative[1][x][0 ]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} 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} \end{align*}

✓ Sympy. Time used: 0.253 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{2} \gamma \left (\frac {1}{3}, \frac {t^{3}}{3}\right ) \]