65.12.6 problem 19.1 (vi)

Internal problem ID [13722]
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 (vi)
Date solved : Wednesday, March 05, 2025 at 10:13:55 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }-3 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 7

ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)-3*y(x) = 0; 
ic:=y(1) = 1, D(y)(1) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {1}{x} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 169

ode=x^2*D[y[x],{x,2}]-x*y[x]-3*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\sqrt {x} \left (\left (3 \operatorname {BesselI}\left (-\sqrt {13},2\right )+\operatorname {BesselI}\left (-1-\sqrt {13},2\right )+\operatorname {BesselI}\left (1-\sqrt {13},2\right )\right ) \operatorname {BesselI}\left (\sqrt {13},2 \sqrt {x}\right )-\left (3 \operatorname {BesselI}\left (\sqrt {13},2\right )+\operatorname {BesselI}\left (-1+\sqrt {13},2\right )+\operatorname {BesselI}\left (1+\sqrt {13},2\right )\right ) \operatorname {BesselI}\left (-\sqrt {13},2 \sqrt {x}\right )\right )}{\operatorname {BesselI}\left (\sqrt {13},2\right ) \left (\operatorname {BesselI}\left (-1-\sqrt {13},2\right )+\operatorname {BesselI}\left (1-\sqrt {13},2\right )\right )-\operatorname {BesselI}\left (-\sqrt {13},2\right ) \left (\operatorname {BesselI}\left (-1+\sqrt {13},2\right )+\operatorname {BesselI}\left (1+\sqrt {13},2\right )\right )} \]

✓ Sympy. Time used: 0.160 (sec). Leaf size: 5

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) - 3*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {1}{x} \]