85.8.3 problem 2 (b)

Internal problem ID [22473]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. C Exercises at page 33
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 08:40:29 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+y^{\prime }+y x&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 22

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

\[ y = -\pi \operatorname {BesselY}\left (1, 1\right ) \operatorname {BesselJ}\left (0, x\right )+\operatorname {BesselJ}\left (1, 1\right ) \pi \operatorname {BesselY}\left (0, x\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {2 \operatorname {BesselJ}(1,1) \operatorname {BesselY}(0,x)-2 \operatorname {BesselY}(1,1) \operatorname {BesselJ}(0,x)}{\operatorname {BesselJ}(1,1) \operatorname {BesselY}(0,1)-\operatorname {BesselJ}(0,1) \operatorname {BesselY}(1,1)} \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \frac {2 J_{0}\left (x\right ) Y_{1}\left (1\right )}{J_{0}\left (1\right ) Y_{1}\left (1\right ) - J_{1}\left (1\right ) Y_{0}\left (1\right )} - \frac {2 J_{1}\left (1\right ) Y_{0}\left (x\right )}{J_{0}\left (1\right ) Y_{1}\left (1\right ) - J_{1}\left (1\right ) Y_{0}\left (1\right )} \]