87.10.3 problem 3
Internal
problem
ID
[23404]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
2.
Linear
differential
equations.
Exercise
at
page
79
Problem
number
:
3
Date
solved
:
Thursday, October 02, 2025 at 09:41:08 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} x y^{\prime \prime }+y&=0 \end{align*}
With initial conditions
\begin{align*}
y \left (1\right )&=1 \\
y^{\prime }\left (1\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 45
ode:=x*diff(diff(y(x),x),x)+y(x) = 0;
ic:=[y(1) = 1, D(y)(1) = 1];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = -\left (\left (\operatorname {BesselY}\left (1, 2\right )-\operatorname {BesselY}\left (0, 2\right )\right ) \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \left (\operatorname {BesselJ}\left (0, 2\right )-\operatorname {BesselJ}\left (1, 2\right )\right )\right ) \sqrt {x}\, \pi
\]
✓ Mathematica. Time used: 0.026 (sec). Leaf size: 87
ode=x*D[y[x],{x,2}]+y[x]==0;
ic={y[1]==1,Derivative[1][y][1] ==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt {x} \left ((\operatorname {BesselY}(0,2)-\operatorname {BesselY}(1,2)-\operatorname {BesselY}(2,2)) \operatorname {BesselJ}\left (1,2 \sqrt {x}\right )+(-\operatorname {BesselJ}(0,2)+\operatorname {BesselJ}(1,2)+\operatorname {BesselJ}(2,2)) \operatorname {BesselY}\left (1,2 \sqrt {x}\right )\right )}{(\operatorname {BesselJ}(2,2)-\operatorname {BesselJ}(0,2)) \operatorname {BesselY}(1,2)+\operatorname {BesselJ}(1,2) (\operatorname {BesselY}(0,2)-\operatorname {BesselY}(2,2))} \end{align*}
✓ Sympy. Time used: 0.099 (sec). Leaf size: 99
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), (x, 2)) + y(x),0)
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): 1}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \sqrt {x} \left (\frac {\left (Y_{2}\left (2\right ) - Y_{0}\left (2\right ) + Y_{1}\left (2\right )\right ) J_{1}\left (2 \sqrt {x}\right )}{J_{1}\left (2\right ) Y_{2}\left (2\right ) - J_{1}\left (2\right ) Y_{0}\left (2\right ) + J_{0}\left (2\right ) Y_{1}\left (2\right ) - J_{2}\left (2\right ) Y_{1}\left (2\right )} + \frac {\left (- J_{1}\left (2\right ) - J_{2}\left (2\right ) + J_{0}\left (2\right )\right ) Y_{1}\left (2 \sqrt {x}\right )}{J_{1}\left (2\right ) Y_{2}\left (2\right ) - J_{1}\left (2\right ) Y_{0}\left (2\right ) + J_{0}\left (2\right ) Y_{1}\left (2\right ) - J_{2}\left (2\right ) Y_{1}\left (2\right )}\right )
\]