67.13.24 problem 20.2 (f)

Internal problem ID [16689]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 20. Euler equations. Additional exercises page 382
Problem number : 20.2 (f)
Date solved : Thursday, October 02, 2025 at 01:37:46 PM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

\begin{align*} y \left (1\right )&=9 \\ y^{\prime }\left (1\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+13*y(x) = 0; 
ic:=[y(1) = 9, D(y)(1) = 3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x^{2} \left (-5 \sin \left (3 \ln \left (x \right )\right )+9 \cos \left (3 \ln \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 24

ode=x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+13*y[x]==0; 
ic={y[1]==9,Derivative[1][y][1]==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 (9 \cos (3 \log (x))-5 \sin (3 \log (x))) \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 3*x*Derivative(y(x), x) + 13*y(x),0) 
ics = {y(1): 9, Subs(Derivative(y(x), x), x, 1): 3} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{2} \left (- 5 \sin {\left (3 \log {\left (x \right )} \right )} + 9 \cos {\left (3 \log {\left (x \right )} \right )}\right ) \]