61.24.70 problem 70

Internal problem ID [12404]
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 : Monday, March 31, 2025 at 05:30:22 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&={\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*}

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

ode:=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); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\tan \left (\frac {\operatorname {RootOf}\left (2 a x \lambda \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, a +b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 a \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}}{2}\right ) a \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}+2 a \lambda x +b \right ) {\mathrm e}^{\lambda x}}{2 \lambda } \]

✗ Mathematica

ode=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); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq((-2*a*lambda_*x - a - b)*y(x)*exp(lambda_*x) + (a**2*lambda_*x**2 + a*b*x + c)*exp(2*lambda_*x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out