Internal problem ID [10751]
Internal file name [OUTPUT/9699_Monday_June_06_2022_03_29_10_PM_72950326/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: 15.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (\left (m -1\right ) x +1\right ) y}{x}=\frac {a^{2} \left (m x +1\right ) \left (x -1\right )}{x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -a^{2} m \,x^{2}-y a m x +a^{2} m x +y y^{\prime } x +a x y-a^{2} x -a y+a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{2} m \,x^{2}+y a m x -a^{2} m x -a x y+a^{2} x +a y-a^{2}}{x y} \end {array} \]
Maple trace
`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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 279
dsolve(y(x)*diff(y(x),x)-a*((m-1)*x+1)*1/x*y(x)=a^2*1/x*(m*x+1)*(x-1),y(x), singsol=all)
\[ -\frac {27 \left (m -1\right ) \left (-54 m^{4} x a \left (m +2\right ) \left (m +\frac {1}{2}\right ) \left (\int _{}^{\frac {9 m \left (\left (m -1\right ) y \left (x \right )+3 \left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right ) a \right )}{\left (m -1\right ) \left (1+2 m \right ) \left (m +2\right ) \left (-y \left (x \right )+a \right )}}\frac {\textit {\_a} {\left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right )}^{\frac {1}{1+m}} {\left (\left (2 m^{2}-m -1\right ) \textit {\_a} +9 m \right )}^{\frac {m}{1+m}}}{8 \left (\left (m^{2}-\frac {1}{2} m -\frac {1}{2}\right ) \textit {\_a} +\frac {9 m}{2}\right ) \left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right ) {\left (\left (m^{2}+\frac {5}{2} m +1\right ) \textit {\_a} +\frac {9 m}{2}\right )}^{2}}d \textit {\_a} \right )+\left (-y \left (x \right )+a \right ) \left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right ) \left (\frac {\left (\left (-1+x \right ) a +y \left (x \right )\right ) m^{2}}{\left (1+2 m \right ) \left (-y \left (x \right )+a \right )}\right )^{\frac {1}{1+m}} \left (\frac {\left (a m x +a -y \left (x \right )\right ) m}{\left (m +2\right ) \left (-y \left (x \right )+a \right )}\right )^{\frac {m}{1+m}}-\frac {2 m x c_{1} a \left (m +2\right ) \left (m +\frac {1}{2}\right )}{27}\right )}{m \left (2 m^{3}+3 m^{2}-3 m -2\right ) a x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((m-1)*x+1)*1/x*y[x]==a^2*1/x*(m*x+1)*(x-1),y[x],x,IncludeSingularSolutions -> True]
Not solved