Internal problem ID [7335]
Book: First order enumerated odes
Section: section 1
Problem number: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime } c -b y^{2}=a x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 75
dsolve(c*diff(y(x),x)=a*x+b*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {b a}{c^{2}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\left (\frac {b a}{c^{2}}\right )^{\frac {1}{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (1, -\left (\frac {b a}{c^{2}}\right )^{\frac {1}{3}} x \right )\right ) c}{b \left (c_{1} \operatorname {AiryAi}\left (-\left (\frac {b a}{c^{2}}\right )^{\frac {1}{3}} x \right )+\operatorname {AiryBi}\left (-\left (\frac {b a}{c^{2}}\right )^{\frac {1}{3}} x \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.21 (sec). Leaf size: 628
DSolve[c*y'[x]==a*x+b*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c \left (x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )} y(x)\to -\frac {c \left (x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )} y(x)\to -\frac {c \left (x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c}} \sqrt {\frac {b}{c}} x^{3/2}\right )} \end{align*}