Internal problem ID [10076]
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: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {\left (y+x a +b \right ) y^{\prime }-\alpha y-\beta x -\gamma =0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 186
dsolve((y(x)+a*x+b)*diff(y(x),x)=alpha*y(x)+beta*x+gamma,y(x), singsol=all)
\[ y \relax (x ) = \frac {a \gamma -b \beta -\frac {\left (x \left (a \alpha -\beta \right )+\alpha b -\gamma \right ) \left (\tan \left (\RootOf \left (\sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \ln \left (-\frac {\left (x \left (a \alpha -\beta \right )+\alpha b -\gamma \right )^{2} \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2}-2 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a \alpha +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \alpha ^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \beta +a^{2}-2 a \alpha +\alpha ^{2}+4 \beta \right )}{4}\right )+2 c_{1} \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }+2 \textit {\_Z} a +2 \textit {\_Z} \alpha \right )\right ) \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }-a +\alpha \right )}{2}}{-a \alpha +\beta } \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y[x]*a*x+b)*y'[x]==\[Alpha]*y[x]+\[Beta]*x+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
Not solved