Internal problem ID [10081]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y x y^{\prime }+n y^{2}-a \left (2 n +1\right ) x y-b y+a^{2} n \,x^{2}+a b x -c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 223
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)
\[ c_{1}+\frac {\left (\frac {-a^{2} n \,x^{2}-x \left (-2 n y \relax (x )+b \right ) a -n y \relax (x )^{2}+b y \relax (x )+c}{\left (x a -y \relax (x )\right )^{2}}\right )^{-\frac {1}{2 n}} \left (\frac {1}{x a -y \relax (x )}\right )^{\frac {1}{n}} y \relax (x ) {\mathrm e}^{\frac {b \arctanh \left (\frac {b \left (x a -y \relax (x )\right )-2 c}{\sqrt {b^{2}+4 c n}\, \left (x a -y \relax (x )\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (\int _{}^{\frac {1}{x a -y \relax (x )}}\left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \arctanh \left (\frac {-2 \textit {\_a} c +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}} \textit {\_a}^{\frac {1}{n}}d \textit {\_a} \right ) a x \left (x a -y \relax (x )\right )}{\left (x a -y \relax (x )\right ) x} = 0 \]
✗ 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