60.2.83 problem 659

Internal problem ID [10657]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 659
Date solved : Wednesday, March 05, 2025 at 12:15:21 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 1.085 (sec). Leaf size: 216

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

\begin{align*} y &= \frac {-a^{2} x^{2}-2 a x b -b^{2}+4 c}{4 a} \\ \frac {\left (1-4 y c_{1} a +c_{1} \left (x^{4}-x^{2}\right ) a^{2}-2 a b c_{1} x +\left (-b^{2}+4 c \right ) c_{1} \right ) \sqrt {a^{2} x^{2}+2 a x b +4 a y+b^{2}-4 c}-x^{2} a \left (-1-4 y c_{1} a +c_{1} \left (x^{4}-x^{2}\right ) a^{2}-2 a b c_{1} x +\left (-b^{2}+4 c \right ) c_{1} \right )}{\left (-4 a y+\left (x^{4}-x^{2}\right ) a^{2}-2 a x b -b^{2}+4 c \right ) \left (a \,x^{2}-\sqrt {a^{2} x^{2}+2 a x b +4 a y+b^{2}-4 c}\right )} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 40.722 (sec). Leaf size: 70

ode=D[y[x],x] == -1/2*b - (a*x)/2 + x*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]
 

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

✓ Sympy. Time used: 1.581 (sec). Leaf size: 32

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*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)
 

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