68.15.59 problem 59

Internal problem ID [17785]
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 : 59
Date solved : Thursday, October 02, 2025 at 02:28:13 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 )&=0 \\ y^{\prime }\left (-1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 27

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

\[ y = -\cosh \left (2 \pi \right ) \sin \left (2 \ln \left (x \right )\right )+i \sinh \left (2 \pi \right ) \cos \left (2 \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to i \sinh (2 (\pi +i \log (x))) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 54

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): 0, Subs(Derivative(y(x), x), x, -1): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {\sin {\left (2 \log {\left (x \right )} \right )} \cosh {\left (2 \pi \right )}}{- \sinh ^{2}{\left (2 \pi \right )} + \cosh ^{2}{\left (2 \pi \right )}} + \frac {i \cos {\left (2 \log {\left (x \right )} \right )} \sinh {\left (2 \pi \right )}}{- \sinh ^{2}{\left (2 \pi \right )} + \cosh ^{2}{\left (2 \pi \right )}} \]