61.24.67 problem 67

Internal problem ID [12480]
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 : 67
Date solved : Tuesday, January 28, 2025 at 08:00:00 PM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 153

dsolve(y(x)*diff(y(x),x)=(a*exp(x)+b)*y(x)+c*exp(2*x)-a*b*exp(x)-b^2,y(x), singsol=all)
 

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

✗ Solution by Mathematica

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

DSolve[y[x]*D[y[x],x]==(a*Exp[x]+b)*y[x]+c*Exp[2*x]-a*b*Exp[x]-b^2,y[x],x,IncludeSingularSolutions -> True]
 

Not solved