Internal problem ID [7336]
Book: First order enumerated odes
Section: section 1
Problem number: 20.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime } c -\frac {x a +b y^{2}}{r}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 91
dsolve(c*diff(y(x),x)=(a*x+b*y(x)^2)/r,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {b a}{r^{2} c^{2}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{\frac {1}{3}} x \right )\right ) r c}{b \left (c_{1} \operatorname {AiryAi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{\frac {1}{3}} x \right )+\operatorname {AiryBi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{\frac {1}{3}} x \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.222 (sec). Leaf size: 517
DSolve[c*y'[x]==(a*x+b*y[x]^2)/r,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )} \\ y(x)\to -\frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )} \\ \end{align*}