87.2.14 problem 16
Internal
problem
ID
[23249]
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
1.
Elementary
methods.
First
order
differential
equations.
Exercise
at
page
17
Problem
number
:
16
Date
solved
:
Sunday, October 12, 2025 at 05:55:08 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=x^{2}+y^{2} \end{align*}
With initial conditions
\begin{align*}
y \left (0\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.349 (sec). Leaf size: 139
ode:=diff(y(x),x) = x^2+y(x)^2;
ic:=[y(0) = 1];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = -\left (\left \{\begin {array}{cc} \frac {\left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right )-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right ) x}{\left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )-\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & x <0 \\ -1 & x =0 \\ \frac {x \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right )+\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right )}{\left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & 0<x \end {array}\right .\right )
\]
✓ Mathematica. Time used: 0.233 (sec). Leaf size: 114
ode=D[y[x],x]==x^2+y[x]^2;
ic={y[0]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\operatorname {Gamma}\left (\frac {3}{4}\right ) \left (x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )\right )-x^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {x^2}{2}\right )}{x \left (\operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {x^2}{2}\right )-2 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {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(0): 1}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list