54.2.319 problem 898

Internal problem ID [12193]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 898
Date solved : Wednesday, October 01, 2025 at 01:08:13 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 95

ode:=diff(y(x),x) = 1/16/x^6*(32*x^5*y(x)+8*x^3+32*x^5+64*x^6*y(x)^3+48*x^4*y(x)^2+12*x^2*y(x)+1)/(4*x^2*y(x)+1+4*x^2); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.519 (sec). Leaf size: 106

ode=D[y[x],x] == (1/16 + x^3/2 + 2*x^5 + (3*x^2*y[x])/4 + 2*x^5*y[x] + 3*x^4*y[x]^2 + 4*x^6*y[x]^3)/(x^6*(1 + 4*x^2 + 4*x^2*y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {256 x^2-\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (-64+\sqrt {\frac {8192}{x}+c_1}\right )}\\ y(x)&\to -\frac {256 x^2+\sqrt {\frac {8192}{x}+c_1}+64}{4 x^2 \left (64+\sqrt {\frac {8192}{x}+c_1}\right )}\\ y(x)&\to -\frac {1}{4 x^2} \end{align*}

✓ Sympy. Time used: 2.322 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (64*x**6*y(x)**3 + 32*x**5*y(x) + 32*x**5 + 48*x**4*y(x)**2 + 8*x**3 + 12*x**2*y(x) + 1)/(16*x**6*(4*x**2*y(x) + 4*x**2 + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x^{2} \left (- C_{1} x - 2 x^{3} + 1\right ) + 2 \sqrt {x^{9} \left (- 2 C_{1} x + x + 2\right )}}{4 x^{4} \left (C_{1} x - 1\right )}, \ y{\left (x \right )} = \frac {- C_{1} x^{3} - 2 x^{5} + x^{2} - 2 \sqrt {x^{9} \left (- 2 C_{1} x + x + 2\right )}}{4 x^{4} \left (C_{1} x - 1\right )}\right ] \]