Internal problem ID [11223]
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.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {a^{2} y {y^{\prime }}^{2}-2 y^{\prime } x +y=0} \]
✓ Solution by Maple
Time used: 0.125 (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 \left (x \right ) &= -\frac {x}{a} \\ y \left (x \right ) &= \frac {x}{a} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\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: 31.661 (sec). Leaf size: 244
DSolve[a^2*y[x]*(y'[x])^2-2*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )-8 i x}}{4 a} \\ y(x)\to \frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )-8 i x}}{4 a} \\ y(x)\to -\frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )+8 i x}}{4 a} \\ y(x)\to \frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )+8 i x}}{4 a} \\ y(x)\to -\frac {x}{a} \\ y(x)\to \frac {x}{a} \\ \end{align*}