69.12.27 problem 301

Internal problem ID [18180]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 301
Date solved : Thursday, October 02, 2025 at 03:07:59 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 53

ode:=2*x^5+4*x^3*y(x)-2*x*y(x)^2+(y(x)^2+2*x^2*y(x)-x^4)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_1}{2}-\frac {\sqrt {-4 x^{4}+4 c_1 \,x^{2}+c_1^{2}}}{2} \\ y &= \frac {c_1}{2}+\frac {\sqrt {-4 x^{4}+4 c_1 \,x^{2}+c_1^{2}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 14.776 (sec). Leaf size: 87

ode=2*(x^5+2*x^3*y[x]-y[x]^2*x)+(y[x]^2+2*x^2*y[x]-x^4)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 15.557 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**5 + 4*x**3*y(x) - 2*x*y(x)**2 + (-x**4 + 2*x**2*y(x) + y(x)**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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