73.4.4 problem 5.1 (d)
Internal
problem
ID
[15015]
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
(d)
Date
solved
:
Monday, March 31, 2025 at 01:12:27 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=1+\left (x y+3 y\right )^{2} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 57
ode:=diff(y(x),x) = 1+(x*y(x)+3*y(x))^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\operatorname {BesselY}\left (-\frac {1}{4}, \frac {\left (x +3\right )^{2}}{2}\right ) c_1 -\operatorname {BesselJ}\left (-\frac {1}{4}, \frac {\left (x +3\right )^{2}}{2}\right )}{\left (\operatorname {BesselY}\left (\frac {3}{4}, \frac {\left (x +3\right )^{2}}{2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {3}{4}, \frac {\left (x +3\right )^{2}}{2}\right )\right ) \left (x +3\right )}
\]
✓ Mathematica. Time used: 0.453 (sec). Leaf size: 351
ode=D[y[x],x]==1+(x*y[x]+3*y[x])^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\left ((x+3)^3\right )^{2/3} \operatorname {Gamma}\left (\frac {7}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )+3 \operatorname {Gamma}\left (\frac {7}{4}\right ) \operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )-\left ((x+3)^3\right )^{2/3} \operatorname {Gamma}\left (\frac {7}{4}\right ) \operatorname {BesselJ}\left (\frac {7}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )+4 c_1 \left ((x+3)^3\right )^{2/3} \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselJ}\left (-\frac {7}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )+12 c_1 \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )-4 c_1 \left ((x+3)^3\right )^{2/3} \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )}{2 (x+3)^3 \left (\operatorname {Gamma}\left (\frac {7}{4}\right ) \operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )+4 c_1 \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )\right )} \\
y(x)\to \frac {-\left ((x+3)^3\right )^{2/3} \operatorname {BesselJ}\left (-\frac {7}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )-3 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )+\left ((x+3)^3\right )^{2/3} \operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )}{2 (x+3)^3 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \left ((x+3)^3\right )^{2/3}\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(x*y(x) + 3*y(x))**2 + Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list