28.1.85 problem 88

Internal problem ID [4391]
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
Date solved : Monday, January 27, 2025 at 09:11:48 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

Solution by Maple

Time used: 0.262 (sec). Leaf size: 68

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 \left (x \right ) &= {\mathrm e}^{x -\frac {1}{2}} \\ y \left (x \right ) &= -{\mathrm e}^{x -\frac {1}{2}} \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {{\mathrm e}^{-2 x +2 c_{1}} \left (-c_{1} +x \right )}\, {\mathrm e}^{x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {{\mathrm e}^{-2 x +2 c_{1}} \left (-c_{1} +x \right )}\, {\mathrm e}^{x} \\ \end{align*}

Solution by Mathematica

Time used: 0.339 (sec). Leaf size: 59

DSolve[2*x*D[y[x],x]-y[x]==D[y[x],x]*Log[y[x]*D[y[x],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*}