28.1.50 problem 51

Internal problem ID [4356]
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 : 51
Date solved : Monday, January 27, 2025 at 09:08:50 AM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 16

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

\[ y \left (x \right ) = -{\mathrm e}^{x} \operatorname {LambertW}\left (-{\mathrm e}^{-x} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 6.758 (sec). Leaf size: 306

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

\[ \text {Solve}\left [\frac {1}{9} 2^{2/3} \left (\frac {\left (\frac {e^x-\frac {3 e^{2 x}}{e^x-y(x)}}{\sqrt [3]{e^{3 x}}}+2\right ) \left (\frac {e^x \left (y(x)+2 e^x\right )}{\sqrt [3]{e^{3 x}} \left (e^x-y(x)\right )}+1\right ) \left (\left (\frac {e^x-\frac {3 e^{2 x}}{e^x-y(x)}}{\sqrt [3]{e^{3 x}}}-1\right ) \log \left (2^{2/3} \left (\frac {e^x-\frac {3 e^{2 x}}{e^x-y(x)}}{\sqrt [3]{e^{3 x}}}+2\right )\right )+\left (\frac {e^x \left (y(x)+2 e^x\right )}{\sqrt [3]{e^{3 x}} \left (e^x-y(x)\right )}+1\right ) \log \left (2^{2/3} \left (\frac {e^x \left (y(x)+2 e^x\right )}{\sqrt [3]{e^{3 x}} \left (e^x-y(x)\right )}+1\right )\right )-3\right )}{\frac {\left (y(x)+2 e^x\right )^3}{\left (e^x-y(x)\right )^3}-\frac {3 e^x \left (y(x)+2 e^x\right )}{\sqrt [3]{e^{3 x}} \left (e^x-y(x)\right )}-2}+e^{-2 x} \left (e^{3 x}\right )^{2/3} x\right )=c_1,y(x)\right ] \]