60.1.323 problem 324

Internal problem ID [10337]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 324
Date solved : Monday, January 27, 2025 at 07:13:25 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 809

dsolve((2*x^3*y(x)^3-x)*diff(y(x),x)+2*x^3*y(x)^3-y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}}{6 x}+\frac {\left (c_{1} -2 x \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}}+\frac {c_{1}}{6}-\frac {x}{3} \\ y &= \frac {-2 \left (-c_{1} x +2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}-i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}} x} \\ y &= \frac {2 \left (c_{1} x -2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}+i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.129 (sec). Leaf size: 672

DSolve[-y[x] + 2*x^3*y[x]^3 + (-x + 2*x^3*y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-2 x^3+c_1 x^2+\frac {x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{6 x^2} \\ y(x)\to \frac {2 x^2 (-2 x+c_1)-\frac {i \left (\sqrt {3}-i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} \\ y(x)\to \frac {2 x^2 (-2 x+c_1)+\frac {i \left (\sqrt {3}+i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} \\ \end{align*}