69.1.135 problem 194
Internal
problem
ID
[14217]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
194
Date
solved
:
Monday, March 31, 2025 at 12:12:58 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=y^{2}+x \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Maple. Time used: 0.204 (sec). Leaf size: 95
ode:=diff(y(x),x) = x+y(x)^2;
ic:=y(0) = 1;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {\left (-2 \,3^{{5}/{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}\right ) \operatorname {AiryAi}\left (1, -x \right )+\left (3 \,3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{{1}/{3}}\right ) \operatorname {AiryBi}\left (1, -x \right )}{\left (-2 \,3^{{5}/{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}\right ) \operatorname {AiryAi}\left (-x \right )+\left (3 \,3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{{1}/{3}}\right ) \operatorname {AiryBi}\left (-x \right )}
\]
✓ Mathematica. Time used: 1.074 (sec). Leaf size: 145
ode=D[y[x],x]==y[x]^2+x;
ic={y[0]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2 x^{3/2}}{3}\right )-x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2 x^{3/2}}{3}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )-2 x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2 x^{3/2}}{3}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2 x^{3/2}}{3}\right )-\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x - 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