84.1.3 problem 1.3
Internal
problem
ID
[22067]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
1.
Basic
concepts
Problem
number
:
1.3
Date
solved
:
Friday, October 03, 2025 at 07:59:45 AM
CAS
classification
:
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} s^{2} t^{\prime \prime }+s t t^{\prime }&=s \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 323
ode:=s^2*diff(diff(t(s),s),s)+s*t(s)*diff(t(s),s) = s;
dsolve(ode,t(s), singsol=all);
\[
t = \frac {-8 \sqrt {-s}\, \sqrt {2}\, \left (c_2 \operatorname {BesselY}\left (\frac {2 \sqrt {2+4 c_1}+\left (2 c_1 +2\right ) \sqrt {2}}{\sqrt {2}+\sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right )-\frac {\operatorname {BesselJ}\left (\frac {2 \sqrt {2+4 c_1}+\left (2 c_1 +2\right ) \sqrt {2}}{\sqrt {2}+\sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right )}{4}\right )+4 \left (\sqrt {2}\, \sqrt {2+4 c_1}+2\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {2+4 c_1}+2+4 c_1 \right )}{2 \sqrt {2}+2 \sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right ) c_2 -\frac {\operatorname {BesselJ}\left (\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {2+4 c_1}+2+4 c_1 \right )}{2 \sqrt {2}+2 \sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right )}{4}\right )}{8 \operatorname {BesselY}\left (\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {2+4 c_1}+2+4 c_1 \right )}{2 \sqrt {2}+2 \sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right ) c_2 -2 \operatorname {BesselJ}\left (\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {2+4 c_1}+2+4 c_1 \right )}{2 \sqrt {2}+2 \sqrt {2+4 c_1}}, \sqrt {2}\, \sqrt {-s}\right )}
\]
✓ Mathematica. Time used: 120.08 (sec). Leaf size: 429
ode=s^2*D[t[s],{s,2}]+s*t[s]*D[t[s],s]==s;
ic={};
DSolve[{ode,ic},t[s],s,IncludeSingularSolutions->True]
\begin{align*} t(s)&\to \frac {\sqrt {2} c_2 \operatorname {Gamma}\left (1-\sqrt {2 c_1+1}\right ) \operatorname {BesselI}\left (-\sqrt {2 c_1+1},\sqrt {2} \sqrt {s}\right )+c_2 \sqrt {s} \operatorname {Gamma}\left (1-\sqrt {2 c_1+1}\right ) \operatorname {BesselI}\left (-\sqrt {2 c_1+1}-1,\sqrt {2} \sqrt {s}\right )+c_2 \sqrt {s} \operatorname {Gamma}\left (1-\sqrt {2 c_1+1}\right ) \operatorname {BesselI}\left (1-\sqrt {2 c_1+1},\sqrt {2} \sqrt {s}\right )+\sqrt {2} i^{2 \sqrt {1+2 c_1}} \operatorname {Gamma}\left (\sqrt {2 c_1+1}+1\right ) \operatorname {BesselI}\left (\sqrt {2 c_1+1},\sqrt {2} \sqrt {s}\right )+i^{2 \sqrt {1+2 c_1}} \sqrt {s} \operatorname {Gamma}\left (\sqrt {2 c_1+1}+1\right ) \operatorname {BesselI}\left (\sqrt {2 c_1+1}-1,\sqrt {2} \sqrt {s}\right )+i^{2 \sqrt {1+2 c_1}} \sqrt {s} \operatorname {Gamma}\left (\sqrt {2 c_1+1}+1\right ) \operatorname {BesselI}\left (\sqrt {2 c_1+1}+1,\sqrt {2} \sqrt {s}\right )}{\sqrt {2} \left (c_2 \operatorname {Gamma}\left (1-\sqrt {2 c_1+1}\right ) \operatorname {BesselI}\left (-\sqrt {2 c_1+1},\sqrt {2} \sqrt {s}\right )+i^{2 \sqrt {1+2 c_1}} \operatorname {Gamma}\left (\sqrt {2 c_1+1}+1\right ) \operatorname {BesselI}\left (\sqrt {2 c_1+1},\sqrt {2} \sqrt {s}\right )\right )} \end{align*}
✗ Sympy
from sympy import *
s = symbols("s")
t = Function("t")
ode = Eq(s**2*Derivative(t(s), (s, 2)) + s*t(s)*Derivative(t(s), s) - s,0)
ics = {}
dsolve(ode,func=t(s),ics=ics)
NotImplementedError : The given ODE -(-s*Derivative(t(s), (s, 2)) + 1)/t(s) + Derivative(t(s), s) cannot be solved by the factorable group method