Internal problem ID [3925]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 678.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {\left (x -x^{2} y-y^{3}\right ) y^{\prime }+y-y^{2} x=x^{3}} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ -\frac {x^{4}}{4}-\frac {y \left (x \right )^{2} x^{2}}{2}+y \left (x \right ) x -\frac {y \left (x \right )^{4}}{4}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.188 (sec). Leaf size: 1807
DSolve[(x-x^2 y[x]-y[x]^3)y'[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*}