Internal problem ID [10203]
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 2.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {4 \,{\mathrm e}^{2 y} \left (y^{\prime }\right )^{2}+2 y^{\prime } x -1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 21
dsolve(4*exp(2*y(x))*diff(y(x),x)^2+2*x*diff(y(x),x)-1=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\ln \left (\frac {1}{4 \,{\mathrm e}^{2 c_{1}}+2 x}\right )}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 9.347 (sec). Leaf size: 119
DSolve[4*Exp[2*y[x]]*(y'[x])^2+2*x*y'[x]-1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right ) \\ y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right ) \\ y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right ) \\ y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \log \left (-\frac {x^2}{4}\right ) \\ \end{align*}