Internal problem ID [7902]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 322.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 y^{4} x +x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve((10*x^2*y(x)^3-3*y(x)^2-2)*diff(y(x),x)+5*x*y(x)^4+x = 0,y(x), singsol=all)
\[ \frac {x^{2} \left (5 y \relax (x )^{4}+1\right )}{2}-y \relax (x )^{3}-2 y \relax (x )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.375 (sec). Leaf size: 2097
DSolve[x + 5*x*y[x]^4 + (-2 - 3*y[x]^2 + 10*x^2*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {3} x^2 \sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}+\sqrt {3} x^2 \sqrt {-\frac {-6 x^2+5 \sqrt [3]{6} x^4 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^4 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}}}{x^6}}-3}{30 x^2} \\ y(x)\to \frac {-\sqrt {3} x^2 \sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}+\sqrt {3} x^2 \sqrt {-\frac {-6 x^2+5 \sqrt [3]{6} x^4 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^4 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}}}{x^6}}+3}{30 x^2} \\ y(x)\to \frac {\sqrt {3} x^2 \sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}-\sqrt {3} x^2 \sqrt {\frac {6 x^2-5 \sqrt [3]{6} x^4 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}-\frac {10\ 6^{2/3} x^4 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}}}{x^6}}+3}{30 x^2} \\ y(x)\to \frac {\sqrt {3} x^2 \sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}+\sqrt {3} x^2 \sqrt {\frac {6 x^2-5 \sqrt [3]{6} x^4 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}-\frac {10\ 6^{2/3} x^4 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} x^2 \sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}+\frac {10\ 6^{2/3} x^2 \left (5 x^4-10 c_1 x^2-2\right )}{\sqrt [3]{189 x^2+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}-18 c_1}}+3}{x^4}}}}{x^6}}+3}{30 x^2} \\ \end{align*}