23.1.719 problem 715

Internal problem ID [5326]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 715
Date solved : Tuesday, September 30, 2025 at 12:30:12 PM
CAS classification : [_rational]

\begin{align*} \left (x +2 y+2 x^{2} y^{3}+x y^{4}\right ) y^{\prime }+\left (1+y^{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 629

ode:=(x+2*y(x)+2*x^2*y(x)^3+x*y(x)^4)*diff(y(x),x)+(1+y(x)^4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-1+\frac {\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+4 c_1 \,x^{4}+18 c_1^{2} x^{2}-x^{2}-4 c_1}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}}{2}-\frac {2 \left (3 c_1 \,x^{2}-1\right )}{\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+4 c_1 \,x^{4}+18 c_1^{2} x^{2}-x^{2}-4 c_1}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}}}{3 c_1 x} \\ y &= \frac {i \left (4-12 c_1 \,x^{2}-\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}}\right ) \sqrt {3}+12 c_1 \,x^{2}-{\left (\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}} x c_1} \\ y &= \frac {12 i \sqrt {3}\, c_1 \,x^{2}+i \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}} \sqrt {3}+12 c_1 \,x^{2}-4 i \sqrt {3}-\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}}-4 \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}-4}{12 c_1 x \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 c_1^{2} x^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 15.385 (sec). Leaf size: 675

ode=(x+2 y[x]+2 x^2 y[x]^3+x y[x]^4)D[y[x],x]+(1+y[x]^4)y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\frac {2 c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x}\\ y(x)&\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to 0\\ y(x)&\to -\sqrt [4]{-1}\\ y(x)&\to \sqrt [4]{-1}\\ y(x)&\to -(-1)^{3/4}\\ y(x)&\to (-1)^{3/4}\\ y(x)&\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right )\\ y(x)&\to -\frac {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**4 + 1)*y(x) + (2*x**2*y(x)**3 + x*y(x)**4 + x + 2*y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out