Internal problem ID [7668]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 88.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 a x}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 420
dsolve(2*diff(y(x),x) - 3*y(x)^2 - 4*a*y(x) - b - c*exp(-2*a*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (-\frac {\sqrt {3}\, \sqrt {c}\, c_{1} \operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )}{3 \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )}-\frac {\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) \sqrt {3}\, \sqrt {c}}{3 \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )}\right ) {\mathrm e}^{-a x}-\frac {\left (\sqrt {4 a^{2}-3 b}\, c_{1} +2 c_{1} a \right ) \operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )+\left (\sqrt {4 a^{2}-3 b}+2 a \right ) \operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )}{3 \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.786 (sec). Leaf size: 1085
DSolve[2*y'[x] - 3*y[x]^2 - 4*a*y[x] - b - c*Exp[-2*a*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-2 a x} \left (3^{\frac {\sqrt {4 a^2-3 b}}{2 a}} a^{\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}} b^{\frac {\sqrt {4 a^2-3 b}}{2 a}} c^{\frac {\sqrt {4 a^2-3 b}}{2 a}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^2-3 b}}{2 a}} \left (\frac {\sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{a}\right )^{\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}} \operatorname {Gamma}\left (\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1\right ) \left (4 a \left (\sqrt {4 a^2-3 b}-2 a\right ) e^{2 a x} \, _0\tilde {F}_1\left (;\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1;-\frac {3 c e^{-2 a x}}{16 a^2}\right )-3 c \, _0\tilde {F}_1\left (;\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+2;-\frac {3 c e^{-2 a x}}{16 a^2}\right )\right )-4^{\frac {\sqrt {4 a^2-3 b}}{a}} a^{\frac {\sqrt {4 a^2-3 b}}{a}} b^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}} c^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}} c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}\right ) \left (4 a \left (2 a+\sqrt {4 a^2-3 b}\right ) e^{2 a x} \, _0\tilde {F}_1\left (;1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2};-\frac {3 c e^{-2 a x}}{16 a^2}\right )+3 c \, _0\tilde {F}_1\left (;2-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2};-\frac {3 c e^{-2 a x}}{16 a^2}\right )\right )\right )}{12 a \left (3^{\frac {\sqrt {4 a^2-3 b}}{2 a}} a^{\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}} b^{\frac {\sqrt {4 a^2-3 b}}{2 a}} c^{\frac {\sqrt {4 a^2-3 b}}{2 a}} \left (\frac {\sqrt {b} \sqrt {c} \sqrt {\frac {e^{-2 a x}}{b}}}{a}\right )^{\frac {\sqrt {4 a^4-3 a^2 b}}{a^2}} \, _0F_1\left (;\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}+1;-\frac {3 c e^{-2 a x}}{16 a^2}\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^2-3 b}}{2 a}}+4^{\frac {\sqrt {4 a^2-3 b}}{a}} a^{\frac {\sqrt {4 a^2-3 b}}{a}} b^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}} c^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}} c_1 \, _0F_1\left (;1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2};-\frac {3 c e^{-2 a x}}{16 a^2}\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2}}\right )} \\ y(x)\to \frac {1}{12} \left (-4 \left (\sqrt {4 a^2-3 b}+2 a\right )-\frac {3 c e^{-2 a x} \, _0\tilde {F}_1\left (;2-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2};-\frac {3 c e^{-2 a x}}{16 a^2}\right )}{a \, _0\tilde {F}_1\left (;1-\frac {\sqrt {4 a^4-3 a^2 b}}{2 a^2};-\frac {3 c e^{-2 a x}}{16 a^2}\right )}\right ) \\ \end{align*}