Internal problem ID [10823]
Internal file name [OUTPUT/9805_Sunday_June_19_2022_09_25_56_PM_53983030/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 1.3.4-2. Equations
of the form \((g_1(x)+g_0(x))y'=f_2(x) y^2+f_1(x) y+f_0(x)\)
Problem number: 7.
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.
\[ \boxed {y y^{\prime } x +y^{2} n -a \left (1+2 n \right ) x y-y b=-n \,a^{2} x^{2}-a b x +c} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } x +y^{2} n -a \left (1+2 n \right ) x y-y b =-n \,a^{2} x^{2}-a b x +c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{2} n +a \left (1+2 n \right ) x y+y b -n \,a^{2} x^{2}-a b x +c}{y x} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 224
dsolve(x*y(x)*diff(y(x),x)=-n*y(x)^2+a*(2*n+1)*x*y(x)+b*y(x)-a^2*n*x^2-a*b*x+c,y(x), singsol=all)
\[ \frac {\left (\frac {-n y \left (x \right )^{2}+\left (2 a x n +b \right ) y \left (x \right )-a^{2} n \,x^{2}-a b x +c}{\left (a x -y \left (x \right )\right )^{2}}\right )^{-\frac {1}{2 n}} \left (\frac {1}{a x -y \left (x \right )}\right )^{\frac {1}{n}} y \left (x \right ) {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-a b x +b y \left (x \right )+2 c}{\sqrt {b^{2}+4 c n}\, \left (-a x +y \left (x \right )\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (\left (\int _{}^{\frac {1}{a x -y \left (x \right )}}\left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-2 c \textit {\_a} +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}} \textit {\_a}^{\frac {1}{n}}d \textit {\_a} \right ) a -c_{1} \right ) \left (a x -y \left (x \right )\right ) x}{x \left (a x -y \left (x \right )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y[x]*y'[x]==-n*y[x]^2+a*(2*n+1)*x*y[x]+b*y[x]-a^2*n*x^2-a*b*x+c,y[x],x,IncludeSingularSolutions -> True]
Not solved