29.25.5 problem 702

Internal problem ID [5292]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 702
Date solved : Monday, January 27, 2025 at 11:04:41 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (2-10 x^{2} y^{3}+3 y^{2}\right ) y^{\prime }&=x \left (1+5 y^{4}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

dsolve((2-10*x^2*y(x)^3+3*y(x)^2)*diff(y(x),x) = x*(1+5*y(x)^4),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.254 (sec). Leaf size: 2097

DSolve[(2-10*x^2*y[x]^3+3*y[x]^2)*D[y[x],x]==x*(1+5*y[x]^4),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*}