68.15.34 problem 34

Internal problem ID [17760]
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 : 34
Date solved : Thursday, October 02, 2025 at 02:27:50 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+2 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: 0.083 (sec). Leaf size: 15

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

\[ y = \sqrt {2}\, \sin \left (\sqrt {2}\, \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 20

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+2*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 \sqrt {2} \sin \left (\sqrt {2} \log (x)\right ) \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + 2*y(x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {2} \sin {\left (\sqrt {2} \log {\left (x \right )} \right )} \]