Internal problem ID [11228]
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]
\[ \boxed {4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }={\mathrm e}^{2 x}} \]
✓ Solution by Maple
Time used: 2.141 (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 \left (x \right ) = \operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{\operatorname {RootOf}\left (\tanh \left (-\frac {\textit {\_Z}}{2}+2+c_{1} -x \right )^{2} {\mathrm e}^{4}+4 \tanh \left (-\frac {\textit {\_Z}}{2}+2+c_{1} -x \right )^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{4}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{2}\right )+c_{1} y \left (x \right ) = -\operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{\operatorname {RootOf}\left (\tanh \left (-\frac {\textit {\_Z}}{2}+2+c_{1} -x \right )^{2} {\mathrm e}^{4}+4 \tanh \left (-\frac {\textit {\_Z}}{2}+2+c_{1} -x \right )^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{4}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{2}\right )+c_{1} \end{align*}
✓ Solution by Mathematica
Time used: 2.772 (sec). Leaf size: 332
DSolve[4*Exp[2*y[x]]*(y'[x])^2+2*Exp[2*x]*y'[x]-Exp[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {2 e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {-\sqrt {4 e^{2 y(x)}+e^{2 x}}+e^x+1}{\sqrt {4 e^{2 y(x)}+e^{2 x}}-e^x+1}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}-\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} y(x)}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+y(x)=c_1,y(x)\right ] \text {Solve}\left [\frac {2 e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {-\sqrt {4 e^{2 y(x)}+e^{2 x}}+e^x+1}{\sqrt {4 e^{2 y(x)}+e^{2 x}}-e^x+1}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} y(x)}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+y(x)=c_1,y(x)\right ] y(x)\to \frac {1}{2} \left (\log \left (-\frac {e^{4 x}}{4}\right )-2 x\right ) \end{align*}