53.3.23 problem 26

Internal problem ID [8485]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 26
Date solved : Monday, January 27, 2025 at 04:07:12 PM
CAS classification : [_rational, _dAlembert]

\begin{align*} 2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 110

dsolve(2*x*diff(y(x),x)^2+(2*x-y(x))*diff(y(x),x)+1-y(x)=0,y(x), singsol=all)
 

\[ y = -2 \,{\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+1\right )} \left (x \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+1\right )}-{\mathrm e}^{2 \operatorname {RootOf}\left (-{\mathrm e}^{3 \textit {\_Z}} x +2 x \,{\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+1\right )} x -\frac {1}{2}\right ) \]

✓ Solution by Mathematica

Time used: 1.433 (sec). Leaf size: 53

DSolve[2*x*(D[y[x],x])^2+(2*x-y[x])*D[y[x],x]+1-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=\frac {\log (K[1]+1)-\frac {K[1]}{K[1]+1}}{K[1]^2}+\frac {c_1}{K[1]^2},y(x)=2 x K[1]+\frac {1}{K[1]+1}\right \},\{y(x),K[1]\}\right ] \]