83.8.32 problem 33

Internal problem ID [19087]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 33
Date solved : Monday, March 31, 2025 at 06:48:08 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }+\frac {a x +b y+c}{b x +f y+e}&=0 \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 85

ode:=diff(y(x),x)+(a*x+b*y(x)+c)/(b*x+f*y(x)+e) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-c_1 a \left (b x +e \right ) f +c_1 \,b^{3} x +c_1 \,b^{2} e -\sqrt {-\left (\left (a x +c \right ) f -b^{2} x -b e \right )^{2} \left (a f -b^{2}\right ) c_1^{2}+f}}{f c_1 \left (a f -b^{2}\right )} \]

✓ Mathematica. Time used: 16.726 (sec). Leaf size: 106

ode=D[y[x],x]+(a*x+b*y[x]+c)/(b*x+f*y[x]+e)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\frac {\sqrt {-x (a x+2 c)+\frac {(b x+e)^2}{f}+c_1 f}}{\sqrt {\frac {1}{f}}}+b x+e}{f} \\ y(x)\to -\frac {b x+e}{f}+\sqrt {\frac {1}{f}} \sqrt {-x (a x+2 c)+\frac {(b x+e)^2}{f}+c_1 f} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
e = symbols("e") 
f = symbols("f") 
y = Function("y") 
ode = Eq((a*x + b*y(x) + c)/(b*x + e + f*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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