62.14.2 problem Ex 2

Internal problem ID [12812]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 25. Equations solvable for \(y\). Page 52
Problem number : Ex 2
Date solved : Monday, March 31, 2025 at 07:10:34 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 4 x {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 35

ode:=4*x*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {x}{4} \\ y &= 4 c_1 +2 \sqrt {c_1 x} \\ y &= 4 c_1 -2 \sqrt {c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 72

ode=4*x*(D[y[x],x])^2+2*x*D[y[x],x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{4} e^{2 c_1} \left (-2 \sqrt {x}+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{4} e^{-4 c_1} \left (1+2 e^{2 c_1} \sqrt {x}\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {x}{4} \\ \end{align*}

✓ Sympy. Time used: 31.485 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x)**2 + 2*x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - \sqrt {C_{1} x}, \ y{\left (x \right )} = C_{1} + \sqrt {C_{1} x}, \ y{\left (x \right )} = - C_{1} - \sqrt {- C_{1} x}, \ y{\left (x \right )} = - C_{1} + \sqrt {- C_{1} x}, \ y{\left (x \right )} = C_{1} - \sqrt {C_{1} x}, \ y{\left (x \right )} = C_{1} + \sqrt {C_{1} x}, \ y{\left (x \right )} = - C_{1} - \sqrt {- C_{1} x}, \ y{\left (x \right )} = - C_{1} + \sqrt {- C_{1} x}\right ] \]