29.29.33 problem 855

Internal problem ID [5438]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 855
Date solved : Monday, January 27, 2025 at 11:23:33 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 50

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

\[ y \left (x \right ) = \left (-\operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )+1\right ) a c_{1} \sqrt {-\frac {x^{2}}{c_{1}^{2} a \operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )}} \]

✓ Solution by Mathematica

Time used: 4.248 (sec). Leaf size: 239

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

\begin{align*} \text {Solve}\left [\frac {i a}{\left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )^2}+\frac {y(x)}{x \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )}+\frac {1}{2} i \log \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\ \text {Solve}\left [-\frac {i a}{\left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )^2}+\frac {y(x)}{x \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )}-\frac {1}{2} i \log \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\ \end{align*}