54.2.73 problem 649

Internal problem ID [11947]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 649
Date solved : Tuesday, September 30, 2025 at 11:46:47 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{4}+\frac {1}{4}+x \sqrt {x^{2}-2 x +1+8 y} \end{align*}

✓ Maple. Time used: 0.087 (sec). Leaf size: 27

ode:=diff(y(x),x) = -1/4*x+1/4+x*(x^2-2*x+1+8*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ c_{1} +\frac {x^{2}}{8}+\frac {17}{128}-\frac {\sqrt {x^{2}-2 x +1+8 y}}{16} = 0 \]

✓ Mathematica. Time used: 0.581 (sec). Leaf size: 36

ode=D[y[x],x] == 1/4 - x/4 + x*Sqrt[1 - 2*x + x^2 + 8*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (4 x^4-(1+16 c_1) x^2+2 x-1+16 c_1{}^2\right ) \end{align*}

✓ Sympy. Time used: 0.802 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sqrt(x**2 - 2*x + 8*y(x) + 1) + x/4 + Derivative(y(x), x) - 1/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x^{2}}{8} + \frac {x}{4} + \frac {\left (C_{1} + x^{2}\right )^{2}}{2} - \frac {1}{8} \]