61.24.70 problem 70

Internal problem ID [12483]
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 : 70
Date solved : Tuesday, January 28, 2025 at 08:00:08 PM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} 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 ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 118

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)
 

\[ y = \frac {\left (\tan \left (\frac {\operatorname {RootOf}\left (2 a x \lambda \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, a +b \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+2 c_{1} a \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}}{2}\right ) a \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}+2 a \lambda x +b \right ) {\mathrm e}^{\lambda x}}{2 \lambda } \]

✗ Solution by Mathematica

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

DSolve[y[x]*D[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],x,IncludeSingularSolutions -> True]
 

Not solved