63.1.55 problem 74

Internal problem ID [15495]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 74
Date solved : Thursday, October 02, 2025 at 10:18:49 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

\begin{align*} \left (y^{3}-x \right ) y^{\prime }&=y \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 18

ode:=(y(x)^3-x)*diff(y(x),x) = y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {c_1}{y}+x -\frac {y^{3}}{4} = 0 \]

✓ Mathematica. Time used: 36.41 (sec). Leaf size: 996

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

\begin{align*} y(x)&\to -\frac {\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}{\sqrt {2} \sqrt [3]{3}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt {2} \sqrt [3]{3} x}{\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 x^2-3 \sqrt {81 x^4-192 c_1{}^3}}}}\\ y(x)&\to \frac {1}{2} \left (\sqrt {-\frac {4 \sqrt {2} \sqrt [3]{3} x}{\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 x^2-3 \sqrt {81 x^4-192 c_1{}^3}}}}-\frac {\sqrt {2} \sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}{\sqrt [3]{3}}\right )\\ y(x)&\to \frac {\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}{\sqrt {2} \sqrt [3]{3}}-\frac {1}{2} \sqrt {\frac {4 \sqrt {2} \sqrt [3]{3} x}{\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 x^2-3 \sqrt {81 x^4-192 c_1{}^3}}}}\\ y(x)&\to \frac {1}{2} \left (\frac {\sqrt {2} \sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}{\sqrt [3]{3}}+\sqrt {\frac {4 \sqrt {2} \sqrt [3]{3} x}{\sqrt {\frac {\left (9 x^2-\sqrt {81 x^4-192 c_1{}^3}\right ){}^{2/3}+4 \sqrt [3]{3} c_1}{\sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}}}-\frac {2 \sqrt [3]{9 x^2-\sqrt {81 x^4-192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{27 x^2-3 \sqrt {81 x^4-192 c_1{}^3}}}}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 37.247 (sec). Leaf size: 648

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2} - \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} - \frac {8 x}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}} - 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2} + \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} - \frac {8 x}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}} - 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2} - \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + \frac {8 x}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}} - 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2} + \frac {\sqrt {\frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + \frac {8 x}{\sqrt {- \frac {8 C_{1}}{3 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}} + 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}} - 2 \sqrt [3]{x^{2} + \sqrt {\frac {64 C_{1}^{3}}{27} + x^{4}}}}}{2}\right ] \]