60.2.85 problem 661

Internal problem ID [10659]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 661
Date solved : Sunday, March 30, 2025 at 06:16:24 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.222 (sec). Leaf size: 39

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

\[ c_{1} +\frac {2 x^{3} a}{3}-\sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c} = 0 \]

✓ Mathematica. Time used: 41.556 (sec). Leaf size: 76

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

\[ y(x)\to -\frac {a^2 x^2+b^2 \left (-\log ^2\left (\sinh \left (\frac {2 a \left (x^3-3 c_1\right )}{3 b}\right )-\cosh \left (\frac {2 a \left (x^3-3 c_1\right )}{3 b}\right )\right )\right )+2 a b x+b^2-4 c}{4 a} \]

✓ Sympy. Time used: 165.725 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \frac {\frac {a^{2} \left (C_{1} + x^{3}\right )^{2}}{9} - \frac {a x \left (a x + 2 b\right )}{4} - \frac {b^{2}}{4} + c}{a} \]