Internal problem ID [10096]
Internal file name [OUTPUT/9043_Monday_June_06_2022_06_16_33_AM_58213099/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1775 (book 6.184).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2}=0} \]
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 checking if the LODE is of Euler type <- LODE of Euler type successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 119
dsolve(a*x^2*y(x)*diff(diff(y(x),x),x)+b*x^2*diff(y(x),x)^2+c*x*y(x)*diff(y(x),x)+d*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= x^{\frac {-\sqrt {\left (-4 a -4 b \right ) d +\left (a -c \right )^{2}}+a -c}{2 a +2 b}} \left (\frac {a^{2}+\left (-2 c -4 d \right ) a -4 b d +c^{2}}{\left (a +b \right )^{2} \left (x^{\frac {\sqrt {\left (-4 a -4 b \right ) d +\left (a -c \right )^{2}}}{a}} c_{1} -c_{2} \right )^{2}}\right )^{-\frac {a}{2 a +2 b}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 61.303 (sec). Leaf size: 92
DSolve[d*y[x]^2 + c*x*y[x]*y'[x] + b*x^2*y'[x]^2 + a*x^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \exp \left (-\frac {\log (x) \left (a \left (\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}-1\right )+c\right )-2 a \log \left (x^{\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}}+c_1\right )}{2 (a+b)}\right ) \]