Internal problem ID [10204]
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 3.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {4 \,{\mathrm e}^{2 y} \left (y^{\prime }\right )^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }-{\mathrm e}^{2 x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 121
dsolve(4*exp(2*y(x))*diff(y(x),x)^2+2*exp(2*x)*diff(y(x),x)-exp(2*x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \arctanh \left (\RootOf \left (-1+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{\RootOf \left (\left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}-x +2\right )\right ) {\mathrm e}^{4}+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}-x +2\right )\right )-{\mathrm e}^{4}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{2}\right )+c_{1} \\ y \relax (x ) = -\arctanh \left (\RootOf \left (-1+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{\RootOf \left (\left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}-x +2\right )\right ) {\mathrm e}^{4}+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}-x +2\right )\right )-{\mathrm e}^{4}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{2}\right )+c_{1} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[4*Exp[2*y[x]]*(y'[x])^2+2*Exp[2*x]*y'[x]-Exp[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
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