60.1.321 problem 322

Internal problem ID [10335]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 322
Date solved : Monday, January 27, 2025 at 07:13:21 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 x y^{4}+x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 29

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)
 

\[ \frac {5 x^{2} y^{4}}{2}+\frac {x^{2}}{2}-y^{3}-2 y+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.252 (sec). Leaf size: 2097

DSolve[x + 5*x*y[x]^4 + (-2 - 3*y[x]^2 + 10*x^2*y[x]^3)*D[y[x],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*}