Internal problem ID [10748]
Internal file name [OUTPUT/9696_Monday_June_06_2022_03_24_54_PM_95434992/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\frac {\left (\left (m +2 L -3\right ) x +n -2 L +3\right ) y}{x}=\left (\left (m -L -1\right ) x^{2}+\left (n -m -2 L +3\right ) x -n +L -2\right ) x^{1-2 L}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & L \,x^{1-2 L} x^{3}-x^{1-2 L} m \,x^{3}+2 L \,x^{1-2 L} x^{2}+x^{1-2 L} m \,x^{2}-x^{1-2 L} n \,x^{2}+x^{1-2 L} x^{3}-2 L y x -L \,x^{1-2 L} x +y y^{\prime } x -y m x +x^{1-2 L} n x -3 x^{1-2 L} x^{2}+2 L y-n y+3 x y+2 x^{1-2 L} x -3 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-L \,x^{1-2 L} x^{3}+x^{1-2 L} m \,x^{3}-2 L \,x^{1-2 L} x^{2}-x^{1-2 L} m \,x^{2}+x^{1-2 L} n \,x^{2}-x^{1-2 L} x^{3}+L \,x^{1-2 L} x +2 L y x -x^{1-2 L} n x +3 x^{1-2 L} x^{2}+y m x -2 L y-2 x^{1-2 L} x +n y-3 x y+3 y}{x y} \end {array} \]
✗ Solution by Maple
dsolve(y(x)*diff(y(x),x)=((m+2*L-3)*x+n-2*L+3)*1/x*y(x)+((m-L-1)*x^2+(n-m-2*L+3)*x-n+L-2)*x^(1-2*L),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==((m+2*L-3)*x+n-2*L+3)*1/x*y[x]+((m-L-1)*x^2+(n-m-2*L+3)*x-n+L-2)*x^(1-2*L),y[x],x,IncludeSingularSolutions -> True]
Timed out