47.3.2 problem 2

Internal problem ID [7475]
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
Date solved : Monday, January 27, 2025 at 03:01:35 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} \frac {y}{x}+\left (y^{3}+\ln \left (x \right )\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 16

dsolve(y(x)/x+(y(x)^3+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ \ln \left (x \right ) y+\frac {y^{4}}{4}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.211 (sec). Leaf size: 1025

DSolve[y[x]/x+(y[x]^3+Log[x])*D[y[x],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 (\frac {\sqrt {\frac {2 \sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-8\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}{\sqrt {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}}}}}}-\frac {\sqrt {\frac {2 \sqrt [3]{3} \left (9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}\right ){}^{2/3}-8\ 3^{2/3} c_1}{\sqrt [3]{9 \log ^2(x)+\sqrt {81 \log ^4(x)+192 c_1{}^3}}}}}{\sqrt {3}}\right ) \\ \end{align*}