73.4.4 problem 5.1 (d)
Internal
problem
ID
[15086]
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
:
Tuesday, January 28, 2025 at 07:30:16 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=1+\left (y x +3 y\right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 57
dsolve(diff(y(x),x)=1+(x*y(x)+3*y(x))^2,y(x), singsol=all)
\[
y = \frac {-\operatorname {BesselY}\left (-\frac {1}{4}, \frac {\left (x +3\right )^{2}}{2}\right ) c_{1} -\operatorname {BesselJ}\left (-\frac {1}{4}, \frac {\left (x +3\right )^{2}}{2}\right )}{\left (\operatorname {BesselY}\left (\frac {3}{4}, \frac {\left (x +3\right )^{2}}{2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {3}{4}, \frac {\left (x +3\right )^{2}}{2}\right )\right ) \left (x +3\right )}
\]
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 351
DSolve[D[y[x],x]==1+(x*y[x]+3*y[x])^2,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*}