Internal problem ID [9209]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1630 (6.40).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-3 y^{\prime } y-3 a y^{2}-4 a^{2} y-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 803
dsolve(diff(diff(y(x),x),x)-3*y(x)*diff(y(x),x)-3*a*y(x)^2-4*a^2*y(x)-b=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}-\frac {6 a^{2}}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+\RootOf \left (2 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a^{2}+3 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a \textit {\_a} +2 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) a^{2}+3 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) a \textit {\_a} +\BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 b a}\, c_{1}+\textit {\_Z} \BesselK \left (\frac {-4 a^{3}+2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a +3 b}{2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1}+\BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 b a}-\textit {\_Z} \BesselI \left (\frac {-4 a^{3}+2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a +3 b}{2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right )^{2}-3 b a}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {6 a^{2}}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+\RootOf \left (2 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a^{2}+3 \BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} a \textit {\_a} +2 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) a^{2}+3 \BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) a \textit {\_a} +\BesselK \left (\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 b a}\, c_{1}+\textit {\_Z} \BesselK \left (\frac {-4 a^{3}+2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a +3 b}{2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1}+\BesselI \left (-\frac {4 a^{3}-3 b}{2 a \sqrt {\left (4 a^{3}-3 b \right ) a}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) \sqrt {4 a^{4}-3 b a}-\textit {\_Z} \BesselI \left (\frac {-4 a^{3}+2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a +3 b}{2 \sqrt {\left (4 a^{3}-3 b \right ) a}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right )^{2}-3 b a}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.521 (sec). Leaf size: 1077
DSolve[-b - 4*a^2*y[x] - 3*a*y[x]^2 - 3*y[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-2 a x} \left (3^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}} b^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} e^{\frac {i \sqrt {4 a^3-3 b} \pi }{2 a^{3/2}}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \left (\frac {\sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ){}^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}} c_1{}^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) \left (2 \sqrt {a} \left (\sqrt {4 a^3-3 b}-2 a^{3/2}\right ) e^{2 a x} \, _0\tilde {F}_1\left (;\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1;\frac {3 e^{-2 a x} c_1}{8 a^2}\right )+3 c_1 \, _0\tilde {F}_1\left (;\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+2;\frac {3 e^{-2 a x} c_1}{8 a^2}\right )\right )-2^{\frac {3 \sqrt {4 a^3-3 b}}{2 a^{3/2}}} a^{\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}} b^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} c_1{}^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} c_2 \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) \left (2 \sqrt {a} \left (2 a^{3/2}+\sqrt {4 a^3-3 b}\right ) e^{2 a x} \, _0\tilde {F}_1\left (;1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3};\frac {3 e^{-2 a x} c_1}{8 a^2}\right )-3 c_1 \, _0\tilde {F}_1\left (;2-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3};\frac {3 e^{-2 a x} c_1}{8 a^2}\right )\right )\right )}{6 a \left (3^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}} b^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} e^{\frac {i \sqrt {4 a^3-3 b} \pi }{2 a^{3/2}}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \left (\frac {\sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ){}^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}} \, _0F_1\left (;\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1;\frac {3 e^{-2 a x} c_1}{8 a^2}\right ) c_1{}^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}}+2^{\frac {3 \sqrt {4 a^3-3 b}}{2 a^{3/2}}} a^{\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}} b^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} c_2 \, _0F_1\left (;1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3};\frac {3 e^{-2 a x} c_1}{8 a^2}\right ) c_1{}^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\right )} \\ \end{align*}