73.4.3 problem 5.1 (c)
Internal
problem
ID
[15085]
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
:
Tuesday, January 28, 2025 at 07:30:14 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }-x y^{2}&=\sqrt {x} \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 45
dsolve(diff(y(x),x)-x*y(x)^2=sqrt(x),y(x), singsol=all)
\[
y = -\frac {\operatorname {BesselY}\left (-\frac {3}{7}, \frac {4 x^{{7}/{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {3}{7}, \frac {4 x^{{7}/{4}}}{7}\right )}{x^{{1}/{4}} \left (\operatorname {BesselY}\left (\frac {4}{7}, \frac {4 x^{{7}/{4}}}{7}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {4}{7}, \frac {4 x^{{7}/{4}}}{7}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.209 (sec). Leaf size: 273
DSolve[D[y[x],x]-x*y[x]^2==Sqrt[x],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*}