Internal problem ID [9995]
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: 1.
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-\left (x a +3 b \right ) y-c \,x^{3}+a \,x^{2} b +2 b^{2} x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 233
dsolve(y(x)*diff(y(x),x)=(a*x+3*b)*y(x)+c*x^3-a*b*x^2-2*b^2*x,y(x), singsol=all)
\[ c_{1}+\frac {\sqrt {-\frac {x^{2}}{b x -y \relax (x )}}\, b \left (b x -y \relax (x )\right ) \left (\int _{}^{-\frac {x^{2}}{b x -y \relax (x )}}\frac {\left (c \,\textit {\_a}^{2}+\textit {\_a} a -2\right )^{\frac {1}{4}} {\mathrm e}^{-\frac {a \arctanh \left (\frac {2 c \textit {\_a} +a}{\sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right )+x \left (\frac {-a \,x^{3} b +c \,x^{4}+a \,x^{2} y \relax (x )-2 b^{2} x^{2}+4 b x y \relax (x )-2 y \relax (x )^{2}}{\left (b x -y \relax (x )\right )^{2}}\right )^{\frac {1}{4}} {\mathrm e}^{-\frac {a \arctanh \left (\frac {-2 c \,x^{2}+a \left (b x -y \relax (x )\right )}{\left (b x -y \relax (x )\right ) \sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}} y \relax (x )}{\sqrt {-\frac {x^{2}}{b x -y \relax (x )}}\, \left (b x -y \relax (x )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*x+3*b)*y[x]+c*x^3-a*b*x^2-2*b^2*x,y[x],x,IncludeSingularSolutions -> True]
Not solved