Internal problem ID [10747]
Internal file name [OUTPUT/9695_Monday_June_06_2022_03_22_44_PM_84902151/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: 11.
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.
\[ \boxed {y y^{\prime }-\left (a \left (2 n +k \right ) x^{2 k}+b \left (2 m -k \right )\right ) x^{m -k -1} y=-\frac {a^{2} m \,x^{4 k}+c \,x^{2 k}+b^{2} m}{x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y x^{2 k} x^{m -k -1} a k x -2 y x^{2 k} x^{m -k -1} a n x +y x^{m -k -1} b k x -2 y x^{m -k -1} b m x +a^{2} m \,x^{4 k}+y y^{\prime } x +b^{2} m +c \,x^{2 k}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y x^{2 k} x^{m -k -1} a k x +2 y x^{2 k} x^{m -k -1} a n x -y x^{m -k -1} b k x +2 y x^{m -k -1} b m x -a^{2} m \,x^{4 k}-b^{2} m -c \,x^{2 k}}{x y} \end {array} \]
✗ Solution by Maple
dsolve(y(x)*diff(y(x),x)=(a*(2*n+k)*x^(2*k)+b*(2*m-k))*x^(m-k-1)*y(x)-(a^2*m*x^(4*k)+c*x^(2*k)+b^2*m)*x^(2*m-2*m-1),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*(2*n+k)*x^(2*k)+b*(2*m-k))*x^(m-k-1)*y[x]-(a^2*m*x^(4*k)+c*x^(2*k)+b^2*m)*x^(2*m-2*m-1),y[x],x,IncludeSingularSolutions -> True]
Timed out