76.18.6 problem Ex. 6

Internal problem ID [20113]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 6
Date solved : Thursday, October 02, 2025 at 05:26:01 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.481 (sec). Leaf size: 143

ode:=a*y(x)*diff(y(x),x)^2+(2*x-b)*diff(y(x),x)-y(x) = 0; 
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: 0.524 (sec). Leaf size: 187

ode=a*y[x]*D[y[x],x]^2+(2*x-b)*D[y[x],x]-y[x]==0; 
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