Internal problem ID [10745]
Internal file name [OUTPUT/9693_Monday_June_06_2022_03_22_23_PM_14690200/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: 9.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-a \left (-b n +x \right ) x^{n -1} y=c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-a \left (-b n +x \right ) x^{n -1} y=c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \left (-b n +x \right ) x^{n -1} y+c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1}}{y} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1972
dsolve(y(x)*diff(y(x),x)=a*(x-n*b)*x^(n-1)*y(x)+c*(x^2-(2*n+1)*b*x+n*(n+1)*b^2)*x^(2*n-1),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.744 (sec). Leaf size: 200
DSolve[y[x]*y'[x]==a*(x-n*b)*x^(n-1)*y[x]+c*(x^2-(2*n+1)*b*x+n*(n+1)*b^2)*x^(2*n-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {a^2 \left (-\frac {2 a \text {arctanh}\left (\frac {a^2-\frac {2 a c (n+1) y(x)}{-b c x^n-b c n x^n+c x^{n+1}}}{a \sqrt {a^2+4 c (n+1)}}\right )}{\sqrt {a^2+4 c (n+1)}}-\log \left (a^2 \left (\frac {a y(x)}{-b c x^n-b c n x^n+c x^{n+1}}+1\right )-\frac {a^2 c (n+1) y(x)^2}{\left (-b c x^n-b c n x^n+c x^{n+1}\right )^2}\right )\right )}{2 c (n+1)}=\frac {a^2 (\log (x-b (n+1))+n \log (x))}{c (n+1)}+c_1,y(x)\right ] \]