60.1.416 problem 418

Internal problem ID [10430]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 418
Date solved : Monday, January 27, 2025 at 07:43:34 PM
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: 0.053 (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 &= 0 \\ y &= -\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: 2.121 (sec). Leaf size: 173

DSolve[a*y[x] - y[x]*D[y[x],x] + x*D[y[x],x]^2==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*}