29.24.16 problem 678

Internal problem ID [5269]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 678
Date solved : Monday, January 27, 2025 at 10:54:24 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 29

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

\[ -\frac {x^{4}}{4}-\frac {x^{2} y \left (x \right )^{2}}{2}+x y \left (x \right )-\frac {y \left (x \right )^{4}}{4}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.206 (sec). Leaf size: 1807

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

\begin{align*} y(x)\to -\frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 x^2}{3}-\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {8 x^2}{3}-\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}-\frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 x^2}{3}+\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}} \\ y(x)\to \frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {-\frac {8 x^2}{3}+\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 (3+4 c_1) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (27-16 c_1{}^2+72 c_1\right ) x^4+64 c_1{}^3}}}} \\ \end{align*}