68.15.33 problem 33

Internal problem ID [17759]
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 : 33
Date solved : Thursday, October 02, 2025 at 02:27:49 PM
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*}

With initial conditions

\begin{align*} y \left (1\right )&=-1 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 11

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(1) = -1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\cos \left (2 \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 12

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==0; 
ic={y[1]==-1,Derivative[1][y][1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\cos (2 \log (x)) \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 10

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 = {y(1): -1, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \cos {\left (2 \log {\left (x \right )} \right )} \]