61.24.1 problem 1

Internal problem ID [12414]
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.
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 07:58:18 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime } y&=\left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 224

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)
 

\[ \frac {x \left (\frac {-2 y^{2}+x \left (a x +4 b \right ) y-a b \,x^{3}+c \,x^{4}-2 b^{2} x^{2}}{\left (b x -y\right )^{2}}\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {-a x b +2 c \,x^{2}+a y}{\left (-b x +y\right ) \sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}} y+\sqrt {\frac {x^{2}}{-b x +y}}\, \left (b x -y\right ) \left (\left (\int _{}^{\frac {x^{2}}{-b x +y}}\frac {\left (\textit {\_a}^{2} c +\textit {\_a} a -2\right )^{{1}/{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 c \textit {\_a} +a}{\sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) b +c_{1} \right )}{\sqrt {\frac {x^{2}}{-b x +y}}\, \left (b x -y\right )} = 0 \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[y[x]*D[y[x],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