60.1.218 problem 223

Internal problem ID [10232]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 223
Date solved : Sunday, March 30, 2025 at 03:33:05 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 51

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

\begin{align*} y &= \frac {c_1 x -\sqrt {5 x^{2} c_1^{2}+4}}{2 c_1} \\ y &= \frac {c_1 x +\sqrt {5 x^{2} c_1^{2}+4}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.46 (sec). Leaf size: 102

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

\begin{align*} y(x)\to \frac {1}{2} \left (x-\sqrt {5 x^2-4 e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x+\sqrt {5 x^2-4 e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x-\sqrt {5} \sqrt {x^2}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {5} \sqrt {x^2}+x\right ) \\ \end{align*}

✓ Sympy. Time used: 1.232 (sec). Leaf size: 34

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

\[ \left [ y{\left (x \right )} = \frac {x}{2} - \frac {\sqrt {C_{1} + 5 x^{2}}}{2}, \ y{\left (x \right )} = \frac {x}{2} + \frac {\sqrt {C_{1} + 5 x^{2}}}{2}\right ] \]