54.2.77 problem 653

Internal problem ID [11951]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 653
Date solved : Tuesday, September 30, 2025 at 11:47:37 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{2}+1+x \sqrt {x^{2}-4 x +4 y} \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 24

ode:=diff(y(x),x) = -1/2*x+1+x*(x^2-4*x+4*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ c_{1} +x^{2}+\frac {1}{4}-\sqrt {x^{2}-4 x +4 y} = 0 \]

✓ Mathematica. Time used: 0.314 (sec). Leaf size: 31

ode=D[y[x],x] == 1 - x/2 + x*Sqrt[-4*x + x^2 + 4*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^4}{4}-\frac {1}{4} (1+4 c_1) x^2+x+c_1{}^2 \end{align*}

✓ Sympy. Time used: 0.753 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sqrt(x**2 - 4*x + 4*y(x)) + x/2 + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x^{2}}{4} + x + \frac {\left (C_{1} + x^{2}\right )^{2}}{4} \]