67.2.29 problem Problem 5(c)
Internal
problem
ID
[13915]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
4,
Second
and
Higher
Order
Linear
Differential
Equations.
Problems
page
221
Problem
number
:
Problem
5(c)
Date
solved
:
Monday, March 31, 2025 at 08:17:57 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}
✓ Maple. Time used: 0.154 (sec). Leaf size: 100
ode:=(x^2+1)*diff(diff(y(x),x),x)+(x-1)*diff(y(x),x)+y(x) = 0;
ic:=y(0) = 0, D(y)(0) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {-20 \,{\mathrm e}^{\left (\frac {1}{4}-\frac {i}{4}\right ) \pi } \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right ) \left (i+x \right )^{\frac {1}{2}+\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {3 i}{2}, \frac {1}{2}-\frac {i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+20 \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {3 i}{2}, \frac {1}{2}-\frac {i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{\left (10-10 i\right ) \left (\operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}\right )-\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {3 i}{2}, \frac {1}{2}-\frac {i}{2}\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}\right )+\left (-1+7 i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}-\frac {i}{2}\right ], \frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {3 i}{2}, \frac {3}{2}-\frac {i}{2}\right ], \left [\frac {5}{2}+\frac {i}{2}\right ], \frac {1}{2}\right )}
\]
✗ Mathematica
ode=(x^2+1)*D[y[x],{x,2}]+(x-1)*D[y[x],x]+y[x]==0;
ic={y[0]==0,Derivative[1][y][0] ==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x - 1)*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + y(x),0)
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1}
dsolve(ode,func=y(x),ics=ics)
False