81.15.7 problem 19-8

Internal problem ID [21715]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 19. Change of variables. Page 483
Problem number : 19-8
Date solved : Thursday, October 02, 2025 at 08:01:07 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} u^{\prime \prime }-3 u^{\prime } x +13 u&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 21

ode:=x^2*diff(diff(u(x),x),x)-3*x*diff(u(x),x)+13*u(x) = 0; 
ic:=[u(1) = -1, D(u)(1) = 1]; 
dsolve([ode,op(ic)],u(x), singsol=all);
 

\[ u = x^{2} \left (\sin \left (3 \ln \left (x \right )\right )-\cos \left (3 \ln \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 22

ode=x^2*D[u[x],{x,2}]-3*x*D[u[x],x]+13*u[x] ==0; 
ic={u[1]==-1,Derivative[1][u][1] ==1}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to x^2 (\sin (3 \log (x))-\cos (3 \log (x))) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
u = Function("u") 
ode = Eq(x**2*Derivative(u(x), (x, 2)) - 3*x*Derivative(u(x), x) + 13*u(x),0) 
ics = {u(1): -1, Subs(Derivative(u(x), x), x, 1): 1} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = x^{2} \left (\sin {\left (3 \log {\left (x \right )} \right )} - \cos {\left (3 \log {\left (x \right )} \right )}\right ) \]