Internal problem ID [2721]
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: 88.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 y^{\prime } x -y-y^{\prime } \ln \left (y y^{\prime }\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 80
dsolve(2*x*diff(y(x),x)-y(x)=diff(y(x),x)*ln(y(x)*diff(y(x),x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = {\mathrm e}^{x -\frac {1}{2}} \\ y \relax (x ) = -{\mathrm e}^{x -\frac {1}{2}} \\ y \relax (x ) = \sqrt {-2 \,{\mathrm e}^{-2 x} {\mathrm e}^{2 c_{1}} c_{1}+2 \,{\mathrm e}^{-2 x} {\mathrm e}^{2 c_{1}} x}\, {\mathrm e}^{x} \\ y \relax (x ) = -\sqrt {-2 \,{\mathrm e}^{-2 x} {\mathrm e}^{2 c_{1}} c_{1}+2 \,{\mathrm e}^{-2 x} {\mathrm e}^{2 c_{1}} x}\, {\mathrm e}^{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.351 (sec). Leaf size: 59
DSolve[2*x*y'[x]-y[x]==y'[x]*Log[y[x]*y'[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{c_1} \sqrt {-2 x+i \pi +2 c_1} \\ y(x)\to e^{c_1} \sqrt {-2 x+i \pi +2 c_1} \\ y(x)\to 0 \\ \end{align*}