2.24.43 Problem 69

Internal problem ID [13607]
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 : 69
Date solved : Friday, December 19, 2025 at 08:25:16 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=\left ({\mathrm e}^{\lambda x} a +b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \\ \end{align*}

Unknown ode type.

2.24.43.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 257

ode:=y(x)*diff(y(x),x) = (a*exp(lambda*x)+b)*y(x)+c*(a^2*exp(2*lambda*x)+a*b*(lambda*x+1)*exp(lambda*x)+b^2*lambda*x); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\sqrt {\left (4 \lambda c +1\right ) \left (3 \lambda c +1\right )^{2}}\, \left (\frac {\lambda c}{2}+\frac {1}{6}\right ) \ln \left (\frac {\left (3 \lambda c +1\right )^{2} \left (b^{2} c \,\lambda ^{2} x^{2}+2 \,{\mathrm e}^{\lambda x} a b c \lambda x +{\mathrm e}^{2 \lambda x} a^{2} c +b \lambda x y+y \,{\mathrm e}^{\lambda x} a -\lambda y^{2}\right ) c}{\left (9 \lambda c +2\right ) y^{2}}\right )-3 \left (\lambda c +\frac {1}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\left (3 \lambda c +1\right ) \left (2 b c \lambda x +2 \,{\mathrm e}^{\lambda x} a c +y\right )}{\sqrt {\left (4 \lambda c +1\right ) \left (3 \lambda c +1\right )^{2}}\, y}\right )+\left (\left (-\lambda c -\frac {1}{3}\right ) \ln \left (\frac {\left (3 \lambda c +1\right ) \left (b \lambda x +a \,{\mathrm e}^{\lambda x}\right ) c}{y}\right )+\left (\lambda c +\frac {1}{3}\right ) \ln \left (b \lambda x +a \,{\mathrm e}^{\lambda x}\right )-c_1 c \lambda \right ) \sqrt {\left (4 \lambda c +1\right ) \left (3 \lambda c +1\right )^{2}}}{\sqrt {\left (4 \lambda c +1\right ) \left (3 \lambda c +1\right )^{2}}\, c \lambda } = 0 \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (a \,{\mathrm e}^{\lambda x}+b \right ) y \left (x \right )+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (a \,{\mathrm e}^{\lambda x}+b \right ) y \left (x \right )+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )}{y \left (x \right )} \end {array} \]

2.24.43.2 ✓ Mathematica. Time used: 0.208 (sec). Leaf size: 134

ode=y[x]*D[y[x],x]==(a*Exp[\[Lambda]*x]+b)*y[x]+c*(a^2*Exp[2*\[Lambda]*x]+a*b*(\[Lambda]*x+1)*Exp[\[Lambda]*x]+b^2*\[Lambda]*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-\frac {\frac {2 \arctan \left (\frac {\frac {2 c \lambda y(x)}{a c e^{\lambda x}+b c \lambda x}-1}{\sqrt {-4 c \lambda -1}}\right )}{\sqrt {-4 c \lambda -1}}+\log \left (-\frac {c \lambda y(x)^2}{\left (a c e^{\lambda x}+b c \lambda x\right )^2}+\frac {y(x)}{a c e^{\lambda x}+b c \lambda x}+1\right )}{2 c \lambda }=\frac {\log \left (a c e^{\lambda x}+b c \lambda x\right )}{c \lambda }+c_1,y(x)\right ] \]

2.24.43.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-c*(a**2*exp(2*lambda_*x) + a*b*(lambda_*x + 1)*exp(lambda_*x) + b**2*lambda_*x) - (a*exp(lambda_*x) + b)*y(x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(a**2*c*exp(2*lambda_*x) + a*b*c*lambda_*x*exp(lambda_*x) + a*b*c*exp(lambda_*x) + b**2*c*lambda_*x + (a*exp(lambda_*x) + b)*y(x))/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method