67.9.27 problem 14.3 (c)

Internal problem ID [16575]
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 (c)
Date solved : Thursday, October 02, 2025 at 01:36:33 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 24

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {3 c_2 \,x^{2}-4 x^{{3}/{2}}+3 c_1}{3 x} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to -\frac {4 \sqrt {x}}{3}+\frac {c_1}{x}+c_2 x \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} + C_{2} x - \frac {4 \sqrt {x}}{3} \]