Internal problem ID [9952]
Internal file name [OUTPUT/8899_Monday_June_06_2022_05_47_53_AM_92975132/index.tex]
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.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }-3 y^{\prime } y-3 a y^{2}-4 a^{2} y=b} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )-3 p \left (y \right ) y +\left (-4 a^{2}-3 a y \right ) y = b \end {align*}
Which is now solved as first order ode for \(p(y)\). Unable to determine ODE type.
Unable to solve. Terminating
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-3*_b(_a)*_a-3*a*_a^2-4*_a*a^2-b = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel Looking for potential symmetries Looking for potential symmetries <- Abel successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 755
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*} -6 a^{2} \left (\int _{}^{y \left (x \right )}\frac {1}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (\left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) \sqrt {4 a^{4}-3 a b}+\left (2 \operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +2 \operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a^{2}+3 \textit {\_a} \left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a +\textit {\_Z} \operatorname {BesselK}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} -\textit {\_Z} \operatorname {BesselI}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right )}^{2}-3 a b}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -6 a^{2} \left (\int _{}^{y \left (x \right )}\frac {1}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (\left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) \sqrt {4 a^{4}-3 a b}+\left (2 \operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +2 \operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a^{2}+3 \textit {\_a} \left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a +\textit {\_Z} \operatorname {BesselK}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} -\textit {\_Z} \operatorname {BesselI}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right )}^{2}-3 a b}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 127.393 (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]
\[ 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 )} \]