2.24.3 Problem 3

Internal problem ID [13567]
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 : 3
Date solved : Friday, December 19, 2025 at 07:04:17 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 y y^{\prime }&=\left (7 x a +5 b \right ) y-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x \\ \end{align*}

Unknown ode type.

2.24.3.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 4589

ode:=2*y(x)*diff(y(x),x) = (7*a*x+5*b)*y(x)-3*a^2*x^3-2*c*x^2-3*b^2*x; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {Expression too large to display} \]

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 
Looking for potential symmetries 
Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (7 a x +5 b \right ) y \left (x \right )-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (7 a x +5 b \right ) y \left (x \right )-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x}{2 y \left (x \right )} \end {array} \]

2.24.3.2 ✗ Mathematica

ode=2*y[x]*D[y[x],x]==(7*a*x+5*b)*y[x]-3*a^2*x^3-2*c*x^2-3*b^2*x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.24.3.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(3*a**2*x**3 + 3*b**2*x + 2*c*x**2 - (7*a*x + 5*b)*y(x) + 2*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0