31.6.20 problem 20

Internal problem ID [5769]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 20
Date solved : Sunday, March 30, 2025 at 10:13:35 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 28.543 (sec). Leaf size: 105

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

\[ \left [ y{\left (x \right )} = C_{1} - 2 \sqrt {C_{1} x}, \ y{\left (x \right )} = C_{1} + 2 \sqrt {C_{1} x}, \ y{\left (x \right )} = - C_{1} - 2 \sqrt {- C_{1} x}, \ y{\left (x \right )} = - C_{1} + 2 \sqrt {- C_{1} x}, \ y{\left (x \right )} = C_{1} - 2 \sqrt {C_{1} x}, \ y{\left (x \right )} = C_{1} + 2 \sqrt {C_{1} x}, \ y{\left (x \right )} = - C_{1} - 2 \sqrt {- C_{1} x}, \ y{\left (x \right )} = - C_{1} + 2 \sqrt {- C_{1} x}\right ] \]