Internal problem ID [5054]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.3. Exact equations problems. page
24
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, [_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {\frac {y}{x}+\left (y^{3}+\ln \relax (x )\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 16
dsolve(y(x)/x+(y(x)^3+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \ln \relax (x ) y \relax (x )+\frac {y \relax (x )^{4}}{4}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 83.193 (sec). Leaf size: 1017
DSolve[y[x]/x+(y[x]^3+Log[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}-\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}-\frac {4 \sqrt {6} \log (x)}{\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}} \\ y(x)\to \frac {1}{2} \left (\sqrt {\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}+\sqrt {\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}-\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}-\frac {4 \sqrt {6} \log (x)}{\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}}\right ) \\ y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}-\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}+\frac {4 \sqrt {6} \log (x)}{\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}} \\ y(x)\to \frac {1}{2} \left (\sqrt {\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}-\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}+\frac {4 \sqrt {6} \log (x)}{\sqrt {\frac {\sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}}-\sqrt {\frac {2 \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}{3^{2/3}}-\frac {8 c_1}{\sqrt [3]{3} \sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}\right ) \\ y(x)\to 0 \\ \end{align*}