69.1.55 problem 74

Internal problem ID [14208]
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 : Tuesday, January 28, 2025 at 06:21:57 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 18

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

\[ -\frac {c_{1}}{y}+x -\frac {y^{3}}{4} = 0 \]

✓ Solution by Mathematica

Time used: 35.568 (sec). Leaf size: 996

DSolve[(y[x]^3-x)*D[y[x],x]==y[x],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*}