73.9.28 problem 14.3 (d)

Internal problem ID [15210]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (d)
Date solved : Thursday, March 13, 2025 at 05:49:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-20 y&=27 x^{5} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{5} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

ode:=x^2*diff(diff(y(x),x),x)-20*y(x) = 27*x^5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {9 \ln \left (x \right ) x^{9}+\left (3 c_{2} -1\right ) x^{9}+3 c_{1}}{3 x^{4}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 29

ode=x^2*D[y[x],{x,2}]-20*y[x]==27*x^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 3 x^5 \log (x)+\left (-\frac {1}{3}+c_2\right ) x^5+\frac {c_1}{x^4} \]

✓ Sympy. Time used: 0.213 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-27*x**5 + x**2*Derivative(y(x), (x, 2)) - 20*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + x^{9} \left (C_{2} + 3 \log {\left (x \right )}\right )}{x^{4}} \]