69.3.1 problem 41
Internal
problem
ID
[17985]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
3.
The
method
of
successive
approximation.
Exercises
page
31
Problem
number
:
41
Date
solved
:
Thursday, October 02, 2025 at 02:32:00 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}
With initial conditions
\begin{align*}
y \left (-1\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.177 (sec). Leaf size: 55
ode:=diff(y(x),x) = x^2-y(x)^2;
ic:=[y(-1) = 0];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselK}\left (\frac {3}{4}, \frac {1}{2}\right )-\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {1}{2}\right )\right )}{\operatorname {BesselK}\left (\frac {3}{4}, \frac {1}{2}\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {1}{2}\right )}
\]
✓ Mathematica. Time used: 0.081 (sec). Leaf size: 211
ode=D[y[x],x]==x^2-y[x]^2;
ic={y[-1]==0};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {i \left (x^2 \left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i}{2}\right )+i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i}{2}\right )+\operatorname {BesselJ}\left (\frac {3}{4},\frac {i}{2}\right )\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+x^2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \left (x^2 \left (-\operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )\right )-i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )\right )}{x \left (2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )+\left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i}{2}\right )+i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i}{2}\right )+\operatorname {BesselJ}\left (\frac {3}{4},\frac {i}{2}\right )\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )\right )} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + y(x)**2 + Derivative(y(x), x),0)
ics = {y(-1): 0}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list