68.15.27 problem 27

Internal problem ID [17753]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 27
Date solved : Thursday, October 02, 2025 at 02:27:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+4 y&=8 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 8; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (2 \ln \left (x \right )\right ) c_2 +\cos \left (2 \ln \left (x \right )\right ) c_1 +2 \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 23

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==8; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x))+2 \end{align*}

✓ Sympy. Time used: 0.161 (sec). Leaf size: 20

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) - 8,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (2 \log {\left (x \right )} \right )} + C_{2} \cos {\left (2 \log {\left (x \right )} \right )} + 2 \]