83.20.2 problem 2

Internal problem ID [19169]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (C) at page 56
Problem number : 2
Date solved : Monday, March 31, 2025 at 06:50:59 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

\begin{align*} \left (2 x -b \right ) y^{\prime }&=y-a y {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.538 (sec). Leaf size: 143

ode:=(2*x-b)*diff(y(x),x) = y(x)-a*y(x)*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {-2 x +b}{2 \sqrt {-a}} \\ y &= \frac {-2 x +b}{2 \sqrt {-a}} \\ y &= 0 \\ y &= \sqrt {\frac {c_1 a +\sqrt {c_1 a \left (-2 x +b \right )^{2}}}{a}} \\ y &= -\sqrt {\frac {c_1 a +\sqrt {c_1 a \left (-2 x +b \right )^{2}}}{a}} \\ y &= \sqrt {-\frac {-c_1 a +\sqrt {c_1 a \left (-2 x +b \right )^{2}}}{a}} \\ y &= -\sqrt {\frac {c_1 a -\sqrt {c_1 a \left (-2 x +b \right )^{2}}}{a}} \\ \end{align*}

✓ Mathematica. Time used: 0.803 (sec). Leaf size: 187

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

\begin{align*} y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to 0 \\ y(x)\to -\frac {i (b-2 x)}{2 \sqrt {a}} \\ y(x)\to \frac {i (b-2 x)}{2 \sqrt {a}} \\ \end{align*}

✗ Sympy

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

Timed Out