29.14.18 problem 399
Internal
problem
ID
[4997]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
14
Problem
number
:
399
Date
solved
:
Monday, January 27, 2025 at 10:02:23 AM
CAS
classification
:
[_rational, [_Riccati, _special]]
\begin{align*} x^{{3}/{2}} y^{\prime }&=a +b \,x^{{3}/{2}} y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 122
dsolve(x^(3/2)*diff(y(x),x) = a+b*x^(3/2)*y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {2 a \left (\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) c_{1} +\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right )\right )}{\sqrt {x}\, \left (2 \sqrt {b}\, \operatorname {BesselJ}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) \sqrt {a}\, x^{{1}/{4}} c_{1} +2 \operatorname {BesselY}\left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}-\operatorname {BesselJ}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right ) c_{1} -\operatorname {BesselY}\left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{{1}/{4}}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.265 (sec). Leaf size: 373
DSolve[x^(3/2) D[y[x],x]==a+ b x^(3/2) y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselY}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-c_1 \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\sqrt {a} \sqrt {b} c_1 \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselY}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-2 b c_1 x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )} \\
y(x)\to -\frac {\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (1,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )+\operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-\sqrt {a} \sqrt {b} \sqrt [4]{x} \operatorname {BesselJ}\left (3,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b x \operatorname {BesselJ}\left (2,4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )} \\
\end{align*}