67.9.28 problem 14.3 (d)

Internal problem ID [16576]
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, October 02, 2025 at 01:36:34 PM
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: 0.004 (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);
 

\[ y = \frac {9 x^{9} \ln \left (x \right )+\left (3 c_2 -1\right ) x^{9}+3 c_1}{3 x^{4}} \]

✓ Mathematica. Time used: 0.01 (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]
 

\begin{align*} y(x)&\to 3 x^5 \log (x)+\left (-\frac {1}{3}+c_2\right ) x^5+\frac {c_1}{x^4} \end{align*}

✓ Sympy. Time used: 0.137 (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)
 

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