Internal problem ID [10040]
Internal file name [OUTPUT/8987_Monday_June_06_2022_06_08_28_AM_60198915/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1719 (book 6.128).
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_xy]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime } y+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{-a +1}=0} \]
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 ) y +\left (a p \left (y \right )+y b \right ) p \left (y \right )+\left (y c +d \,y^{-a}\right ) y = 0 \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 a quadrature checking if the LODE has constant coefficients <- constant coefficients successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 129
dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)*diff(y(x),x)+c*y(x)^2+d*y(x)^(1-a)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (b -\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2 a +2}} \left (\frac {\left (-4 a -4\right ) c^{3}+b^{2} c^{2}}{\left (-d \,{\mathrm e}^{-\frac {\left (-b +\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2}} \sqrt {\left (-4 a -4\right ) c +b^{2}}+\left (a +1\right ) c \left (c_{2} {\mathrm e}^{-x \sqrt {\left (-4 a -4\right ) c +b^{2}}}-c_{1} \right )\right )^{2}}\right )^{-\frac {1}{2 a +2}} \]
✓ Solution by Mathematica
Time used: 61.36 (sec). Leaf size: 396
DSolve[c*y[x]^2 + d*y[x]^(1 - a) + b*y[x]*y'[x] + a*y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\frac {\exp \left (-\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right ) \left (b^2 \left (d e^{\frac {1}{2} x \left (\sqrt {b^2-4 (a+1) c}+b\right )}-c c_2 \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )\right )+(a+1) c \left (-4 d e^{\frac {1}{2} x \left (\sqrt {b^2-4 (a+1) c}+b\right )}+4 c c_2 \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )-c_1 \sqrt {b^2-4 (a+1) c}\right )+b d \sqrt {b^2-4 (a+1) c} e^{\frac {1}{2} x \left (\sqrt {b^2-4 (a+1) c}+b\right )}-b c \left (c_2 \sqrt {b^2-4 (a+1) c} \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+(a+1) c_1\right )\right )}{c \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}\right ){}^{\frac {1}{a+1}} \]