Internal problem ID [6585]
Book: First order enumerated odes
Section: section 1
Problem number: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {c y^{\prime }-\frac {x a +b y^{2}}{r \,x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 106
dsolve(c*diff(y(x),x)=(a*x+b*y(x)^2)/(r*x^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {a \left (\BesselY \left (0, 2 \sqrt {\frac {b a}{r^{2} c^{2} x}}\right ) c_{1}+\BesselJ \left (0, 2 \sqrt {\frac {b a}{r^{2} c^{2} x}}\right )\right )}{c r \sqrt {\frac {b a}{r^{2} c^{2} x}}\, \left (c_{1} \BesselY \left (1, 2 \sqrt {\frac {b a}{r^{2} c^{2} x}}\right )+\BesselJ \left (1, 2 \sqrt {\frac {b a}{r^{2} c^{2} x}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.29 (sec). Leaf size: 179
DSolve[c*y'[x]==(a*x+b*y[x]^2)/(r*x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} c r x \left (c_1 \, _0\tilde {F}_1\left (;1;-\frac {a b}{c^2 r^2 x}\right )+2 i Y_0\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )\right )}{\sqrt {a} b c_1 \, _0\tilde {F}_1\left (;2;-\frac {a b}{c^2 r^2 x}\right )+\frac {2 i \sqrt {b} c r Y_1\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )}{\sqrt {\frac {1}{x}}}} \\ y(x)\to \frac {c r x \, _0\tilde {F}_1\left (;1;-\frac {a b}{c^2 r^2 x}\right )}{b \, _0\tilde {F}_1\left (;2;-\frac {a b}{c^2 r^2 x}\right )} \\ \end{align*}