83.10.11 problem 11

Internal problem ID [21980]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 11
Date solved : Thursday, October 02, 2025 at 08:21:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 35

ode:=y(x) = 4*x*diff(y(x),x)^2+2*x*diff(y(x),x); 
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: 0.086 (sec). Leaf size: 72

ode=y[x]==4*x*D[y[x],x]^2+2*x*D[y[x],x]; 
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: 17.825 (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)
 

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