29.3.4 problem 58
Internal
problem
ID
[4666]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
3
Problem
number
:
58
Date
solved
:
Monday, January 27, 2025 at 09:30:48 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=a \,x^{2}+b y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 73
dsolve(diff(y(x),x) = a*x^2+b*y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {\left (-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) c_{1} -\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right ) \sqrt {a b}\, x}{b \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.171 (sec). Leaf size: 305
DSolve[D[y[x],x]==a x^2+b y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {b} x^2 \left (-2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )} \\
y(x)\to -\frac {\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-\sqrt {a} \sqrt {b} x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )} \\
\end{align*}