1.409 problem 410

Internal problem ID [7990]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 410.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [_rational, _dAlembert]

Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+4 y^{\prime }-2 y=0} \end {gather*}

Solution by Maple

Time used: 0.205 (sec). Leaf size: 64

dsolve(x*diff(y(x),x)^2+4*diff(y(x),x)-2*y(x) = 0,y(x), singsol=all)
 

\[ y \relax (x ) = \frac {x \,{\mathrm e}^{2 \RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_{1}+8 \textit {\_Z} -4 x \right )}}{2}+2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_{1}+8 \textit {\_Z} -4 x \right )} \]

Solution by Mathematica

Time used: 30.555 (sec). Leaf size: 90

DSolve[-2*y[x] + 4*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=-\frac {2 (2 K[1]-y(K[1]))}{K[1]^2},y(x)=4 \left (\frac {2}{K[1]}+\log (K[1])\right ) \exp \left (-4 \left (\frac {1}{2} \log (2-K[1])-\frac {1}{2} \log (K[1])\right )\right )+c_1 \exp \left (-4 \left (\frac {1}{2} \log (2-K[1])-\frac {1}{2} \log (K[1])\right )\right )\right \},\{y(x),K[1]\}\right ] \]