83.19.5 problem 5

Internal problem ID [19233]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (B) at page 55
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 01:15:10 PM
CAS classification : [_rational, _dAlembert]

\begin{align*} y&=x {y^{\prime }}^{2}+y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

Time used: 0.845 (sec). Leaf size: 46

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

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