65.12.11 problem 19.2

Internal problem ID [13727]
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 : 19.2
Date solved : Wednesday, March 05, 2025 at 10:14:12 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} a y^{\prime \prime }+\left (b -a \right ) y^{\prime }+c y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

ode:=a*diff(diff(y(z),z),z)+(b-a)*diff(y(z),z)+c*y(z) = 0; 
dsolve(ode,y(z), singsol=all);
 

\[ y \left (z \right ) = c_{1} {\mathrm e}^{\frac {\left (a -b +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}\right ) z}{2 a}}+{\mathrm e}^{-\frac {\left (-a +b +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}\right ) z}{2 a}} c_{2} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 72

ode=a*D[y[z],{z,2}]+(b-a)*D[y[z],z]+c*y[z]==0; 
ic={}; 
DSolve[{ode,ic},y[z],z,IncludeSingularSolutions->True]
 

\[ y(z)\to \left (c_2 e^{\frac {z \sqrt {a^2-2 a (b+2 c)+b^2}}{a}}+c_1\right ) \exp \left (-\frac {z \left (\sqrt {a^2-2 a (b+2 c)+b^2}-a+b\right )}{2 a}\right ) \]

✓ Sympy. Time used: 0.259 (sec). Leaf size: 70

from sympy import * 
z = symbols("z") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*Derivative(y(z), (z, 2)) + c*y(z) + (-a + b)*Derivative(y(z), z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

\[ y{\left (z \right )} = C_{1} e^{\frac {z \left (1 - \frac {b}{a} - \frac {\sqrt {a^{2} - 2 a b - 4 a c + b^{2}}}{a}\right )}{2}} + C_{2} e^{\frac {z \left (1 - \frac {b}{a} + \frac {\sqrt {a^{2} - 2 a b - 4 a c + b^{2}}}{a}\right )}{2}} \]