Internal problem ID [10064]
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: 70.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-{\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y+{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 122
dsolve(y(x)*diff(y(x),x)=exp(lambda*x)*(2*a*lambda*x+a+b)*y(x)-exp(2*lambda*x)*(a^2*lambda*x^2+a*b*x+c),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\tan \left (\frac {\RootOf \left (2 \,{\mathrm e}^{\textit {\_Z}} a \lambda x +\left (\int _{}^{-\textit {\_Z}}\tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, a +{\mathrm e}^{\textit {\_Z}} b +2 a c_{1}\right ) \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) a \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}+2 a x \lambda +b \right ) {\mathrm e}^{\lambda x}}{2 \lambda } \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==Exp[\[Lambda]*x]*(2*a*\[Lambda]*x+a+b)*y[x]-Exp[2*\[Lambda]*x]*(a^2*\[Lambda]*x^2+a*b*x+c),y[x],x,IncludeSingularSolutions -> True]
Not solved