62.16.3 problem Ex 3

Internal problem ID [12894]
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
Date solved : Tuesday, January 28, 2025 at 04:32:20 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} 4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }-{\mathrm e}^{2 x}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.867 (sec). Leaf size: 43

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)
 

\[ y = \operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{\operatorname {RootOf}\left (-4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x +2\right )^{2}+{\mathrm e}^{4}\right )}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{2}\right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.894 (sec). Leaf size: 176

DSolve[4*Exp[2*y[x]]*(D[y[x],x])^2+2*Exp[2*x]*D[y[x],x]-Exp[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [y(x)-\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\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*}