28.1.31 problem 31

Internal problem ID [4337]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 31
Date solved : Monday, January 27, 2025 at 09:06:02 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y+x \left (y^{2}+\ln \left (x \right )\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 233

dsolve((y(x))+x*(y(x)^2+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{2 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {i \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{2}/{3}} \sqrt {3}+4 i \ln \left (x \right ) \sqrt {3}+\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}-4 \ln \left (x \right )}{4 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {i \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{2}/{3}} \sqrt {3}+4 i \ln \left (x \right ) \sqrt {3}-\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{2}/{3}}+4 \ln \left (x \right )}{4 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.208 (sec). Leaf size: 272

DSolve[(y[x])+x*(y[x]^2+Log[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} \log (x)}{\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) \log (x)+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \log (x)}{2^{2/3} \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}