29.29.35 problem 858

Internal problem ID [5440]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 858
Date solved : Monday, January 27, 2025 at 11:23:41 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 2.599 (sec). Leaf size: 173

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

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