78.11.4 problem 4

Internal problem ID [18196]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 16. The Use of a Known Solution to find Another. Problems at page 121
Problem number : 4
Date solved : Thursday, March 13, 2025 at 11:48:51 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \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)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} x^{4}+c_{2}}{x^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 18

ode=x^2*D[y[x],{x,2}] +x*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 x^4+c_1}{x^2} \]

✓ Sympy. Time used: 0.152 (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) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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