Internal problem ID [9205]
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]]
\[ \boxed {y^{\prime \prime }-3 y^{\prime } y-3 a y^{2}-4 a^{2} y-b=0} \]
✓ Solution by Maple
Time used: 0.25 (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 \left (x \right )}-\frac {6 a^{2}}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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} +\operatorname {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} \operatorname {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} +\operatorname {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} \operatorname {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 \left (x \right )}-\frac {6 a^{2}}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (2 \operatorname {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 \operatorname {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} +\operatorname {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} +2 \operatorname {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 \operatorname {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} +\textit {\_Z} \operatorname {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} +\sqrt {4 a^{4}-3 b a}\, \operatorname {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 )-\textit {\_Z} \operatorname {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: 4.641 (sec). Leaf size: 1670
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 {\sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1} \left (-i^{\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}} 2^{\frac {3 \sqrt {4 a^6-3 a^3 b}}{2 a^3}+\frac {1}{2}} 3^{\frac {1}{2} \left (\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}-1\right )} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}} b^{\frac {1}{2} \left (\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}-1\right )} \left (2 a^3-\sqrt {4 a^3-3 b} a^{3/2}+\sqrt {4 a^6-3 a^3 b}\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {1}{2} \left (\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}-1\right )} \operatorname {BesselI}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3},\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) \operatorname {Gamma}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) c_1{}^{\frac {1}{2} \left (\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}-1\right )}+i^{\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}} 2^{\frac {3 \sqrt {4 a^6-3 a^3 b}}{2 a^3}} 3^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{a^3}+2} b^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \left (\operatorname {BesselI}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1,\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+\operatorname {BesselI}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1,\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) \operatorname {Gamma}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\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}}} 3^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {1}{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}-\frac {1}{2}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {1}{2}} \left (\sqrt {3} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \left (\operatorname {BesselI}\left (-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1,\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+\operatorname {BesselI}\left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3},\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) \sqrt {c_1} a^2+\sqrt {2} \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}+\sqrt {4 a^6-3 a^3 b}\right ) \operatorname {BesselI}\left (-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3},\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) c_2 \operatorname {Gamma}\left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) c_1{}^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {1}{2}}\right )}{\sqrt {6} a^2 \left (i^{\frac {\sqrt {4 a^3-3 b}}{a^{3/2}}} 2^{\frac {3 \sqrt {4 a^6-3 a^3 b}}{2 a^3}} 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}}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {\sqrt {4 a^3-3 b}}{2 a^{3/2}}} \operatorname {BesselI}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3},\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) \operatorname {Gamma}\left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\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}}} 3^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}} 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}} \operatorname {BesselI}\left (-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3},\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_2 \operatorname {Gamma}\left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) c_1{}^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\right )} \\ \end{align*}