60.2.319 problem 897

Internal problem ID [10893]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 897
Date solved : Sunday, March 30, 2025 at 07:20:35 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (-108 x^{{3}/{2}} y+18 x^{{9}/{2}}-108 x^{{3}/{2}}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 95

ode:=diff(y(x),x) = (-108*x^(3/2)*y(x)+18*x^(9/2)-108*x^(3/2)-216*y(x)^3+108*x^3*y(x)^2-18*y(x)*x^6+x^9)*x^(1/2)/(-216*y(x)+36*x^3-216); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3}}{6} \\ y &= \frac {\sqrt {9 c_1 -12 x^{{3}/{2}}}\, x^{3}-3 x^{3}+18}{6 \sqrt {9 c_1 -12 x^{{3}/{2}}}-18} \\ y &= \frac {\sqrt {9 c_1 -12 x^{{3}/{2}}}\, x^{3}+3 x^{3}-18}{6 \sqrt {9 c_1 -12 x^{{3}/{2}}}+18} \\ \end{align*}

✓ Mathematica. Time used: 2.076 (sec). Leaf size: 76

ode=D[y[x],x] == (Sqrt[x]*(-108*x^(3/2) + 18*x^(9/2) + x^9 - 108*x^(3/2)*y[x] - 18*x^6*y[x] + 108*x^3*y[x]^2 - 216*y[x]^3))/(-216 + 36*x^3 - 216*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {x^3}{6}-\frac {216}{216+\sqrt {-62208 x^{3/2}+c_1}} \\ y(x)\to \frac {x^3}{6}+\frac {216}{-216+\sqrt {-62208 x^{3/2}+c_1}} \\ y(x)\to \frac {x^3}{6} \\ \end{align*}

✓ Sympy. Time used: 3.831 (sec). Leaf size: 99

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x)*(18*x**(9/2) - 108*x**(3/2)*y(x) - 108*x**(3/2) + x**9 - 18*x**6*y(x) + 108*x**3*y(x)**2 - 216*y(x)**3)/(36*x**3 - 216*y(x) - 216) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {3 C_{1} x^{3} + 2 x^{\frac {9}{2}} - 3 \sqrt {3} \sqrt {- 6 C_{1} - 4 x^{\frac {3}{2}} + 3} - 9}{6 \left (3 C_{1} + 2 x^{\frac {3}{2}}\right )}, \ y{\left (x \right )} = \frac {3 C_{1} x^{3} + 2 x^{\frac {9}{2}} + 3 \sqrt {3} \sqrt {- 6 C_{1} - 4 x^{\frac {3}{2}} + 3} - 9}{6 \left (3 C_{1} + 2 x^{\frac {3}{2}}\right )}\right ] \]