Internal problem ID [10834]
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`]]
\[ \boxed {x y y^{\prime }+y^{2} n -a \left (2 n +1\right ) x y-b y=-a^{2} n \,x^{2}-a b x +c} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ c_{1} +\frac {\left (\frac {1}{a x -y \left (x \right )}\right )^{\frac {1}{n}} \left (\frac {-n \,a^{2} x^{2}-x \left (-2 y \left (x \right ) n +b \right ) a -n y \left (x \right )^{2}+b y \left (x \right )+c}{\left (a x -y \left (x \right )\right )^{2}}\right )^{-\frac {1}{2 n}} y \left (x \right ) {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {b \left (a x -y \left (x \right )\right )-2 c}{\sqrt {b^{2}+4 c n}\, \left (a x -y \left (x \right )\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (\int _{}^{\frac {1}{a x -y \left (x \right )}}\textit {\_a}^{\frac {1}{n}} \left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-2 \textit {\_a} c +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}}d \textit {\_a} \right ) a x \left (a x -y \left (x \right )\right )}{\left (a x -y \left (x \right )\right ) x} = 0 \]
✗ 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