9.1.1 problem problem 38

Internal problem ID [928]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.2, Higher-Order Linear Differential Equations. General solutions of Linear Equations. Page 288
Problem number : problem 38
Date solved : Saturday, March 29, 2025 at 10:34:52 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-9 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-9*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \,x^{6}+c_1}{x^{3}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 18

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

\[ y(x)\to \frac {c_2 x^6+c_1}{x^3} \]

✓ Sympy. Time used: 0.143 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {C_{1}}{x^{3}} + C_{2} x^{3} \]