67.13.21 problem 20.2 (c)

Internal problem ID [16686]
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 (c)
Date solved : Thursday, October 02, 2025 at 01:37:42 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-11 x y^{\prime }+36 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&={\frac {1}{2}} \\ y^{\prime }\left (1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 14

ode:=x^2*diff(diff(y(x),x),x)-11*x*diff(y(x),x)+36*y(x) = 0; 
ic:=[y(1) = 1/2, D(y)(1) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x^{6} \left (\frac {1}{2}-\ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

ode=x^2*D[y[x],{x,2}]-11*x*D[y[x],x]+36*y[x]==0; 
ic={y[1]==1/2,Derivative[1][y][1]==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} x^6 (1-2 \log (x)) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 11*x*Derivative(y(x), x) + 36*y(x),0) 
ics = {y(1): 1/2, Subs(Derivative(y(x), x), x, 1): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{6} \left (\frac {1}{2} - \log {\left (x \right )}\right ) \]