57.1.21 problem 21

Internal problem ID [9005]
Book : First order enumerated odes
Section : section 1
Problem number : 21
Date solved : Monday, January 27, 2025 at 05:26:55 PM
CAS classification : [_rational, _Riccati]

\begin{align*} c y^{\prime }&=\frac {a x +b y^{2}}{r x} \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 94

dsolve(c*diff(y(x),x)=(a*x+b*y(x)^2)/(r*x),y(x), singsol=all)
 

\[ y = \frac {\sqrt {\frac {x b a}{r^{2} c^{2}}}\, c r \left (\operatorname {BesselY}\left (1, 2 \sqrt {\frac {x b a}{r^{2} c^{2}}}\right ) c_{1} +\operatorname {BesselJ}\left (1, 2 \sqrt {\frac {x b a}{r^{2} c^{2}}}\right )\right )}{b \left (c_{1} \operatorname {BesselY}\left (0, 2 \sqrt {\frac {x b a}{r^{2} c^{2}}}\right )+\operatorname {BesselJ}\left (0, 2 \sqrt {\frac {x b a}{r^{2} c^{2}}}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.291 (sec). Leaf size: 207

DSolve[c*D[y[x],x]==(a*x+b*y[x]^2)/(r*x),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {x} \left (2 \operatorname {BesselY}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )+c_1 \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )\right )}{\sqrt {b} \left (2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )+c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )\right )} \\ y(x)\to \frac {\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )}{\sqrt {b} \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {x}}{c r}\right )} \\ \end{align*}