Internal problem ID [10003]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-a \left (-b n +x \right ) x^{n -1} y-c \left (x^{2}-\left (2 n +1\right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 10211
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: 0.455 (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]
\[ \text {Solve}\left [\frac {a^2 \left (-\log \left (a^2 \left (\frac {a y(x)}{-b c x^n-b c n x^n+c x^{n+1}}+1\right )-\frac {a^2 c (n+1) y(x)^2}{\left (-b c x^n-b c n x^n+c x^{n+1}\right )^2}\right )-\frac {2 a \tanh ^{-1}\left (\frac {a^2-\frac {2 a c (n+1) y(x)}{-b c x^n-b c n x^n+c x^{n+1}}}{a \sqrt {a^2+4 c (n+1)}}\right )}{\sqrt {a^2+4 c (n+1)}}\right )}{2 c (n+1)}=\frac {a^2 (\log (x-b (n+1))+n \log (x))}{c (n+1)}+c_1,y(x)\right ] \]