67.4.3 problem 5.1 (c)
Internal
problem
ID
[16372]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
5.
LINEAR
FIRST
ORDER
EQUATIONS.
Additional
exercises.
page
103
Problem
number
:
5.1
(c)
Date
solved
:
Friday, October 03, 2025 at 12:37:41 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }-x y^{2}&=\sqrt {x} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 45
ode:=diff(y(x),x)-x*y(x)^2 = x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{{7}/{4}}}{7}\right ) c_1 +\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{{7}/{4}}}{7}\right )}{x^{{1}/{4}} \left (\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{{7}/{4}}}{7}\right ) c_1 +\operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{{7}/{4}}}{7}\right )\right )}
\]
✓ Mathematica. Time used: 0.134 (sec). Leaf size: 273
ode=D[y[x],x]-x*y[x]^2==Sqrt[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (-\frac {3}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {11}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-c_1 x^{7/4} \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \left (\operatorname {Gamma}\left (\frac {11}{7}\right ) \operatorname {BesselJ}\left (\frac {4}{7},\frac {4 x^{7/4}}{7}\right )+c_1 \operatorname {Gamma}\left (\frac {3}{7}\right ) \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )\right )}\\ y(x)&\to -\frac {x^{7/4} \operatorname {BesselJ}\left (-\frac {11}{7},\frac {4 x^{7/4}}{7}\right )-x^{7/4} \operatorname {BesselJ}\left (\frac {3}{7},\frac {4 x^{7/4}}{7}\right )+2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )}{2 x^2 \operatorname {BesselJ}\left (-\frac {4}{7},\frac {4 x^{7/4}}{7}\right )} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sqrt(x) - x*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(x) - x*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method