62.16.8 problem Ex 8

Internal problem ID [12899]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 27. Clairaut equation. Page 56
Problem number : Ex 8
Date solved : Tuesday, January 28, 2025 at 04:39:25 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.148 (sec). Leaf size: 51

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

\begin{align*} y &= -\frac {x}{a} \\ y &= \frac {x}{a} \\ y &= 0 \\ y &= {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\textit {\_Z} +c_{1} -\ln \left (x \right )\right )^{2} a^{2}+1\right )} x \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.724 (sec). Leaf size: 89

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

\begin{align*} y(x)\to -\frac {\sqrt {-e^{a^2 c_1} \left (-2 x+e^{a^2 c_1}\right )}}{a} \\ y(x)\to \frac {\sqrt {-e^{a^2 c_1} \left (-2 x+e^{a^2 c_1}\right )}}{a} \\ y(x)\to -\frac {x}{a} \\ y(x)\to \frac {x}{a} \\ \end{align*}