40.5.10 problem 26

Internal problem ID [6675]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 26
Date solved : Monday, January 27, 2025 at 02:18:59 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}-y^{\prime } y-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 32

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

\begin{align*} y &= 0 \\ y &= \frac {\left (\operatorname {LambertW}\left (\frac {x \,{\mathrm e}}{c_1}\right )-1\right )^{2} x}{\operatorname {LambertW}\left (\frac {x \,{\mathrm e}}{c_1}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.342 (sec). Leaf size: 161

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

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}}}\right )+\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}+4}-\sqrt {\frac {y(x)}{x}}}&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}}}\right )-\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}}+\sqrt {\frac {y(x)}{x}+4}}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}