60.1.408 problem 410

Internal problem ID [10422]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 410
Date solved : Monday, January 27, 2025 at 07:43:03 PM
CAS classification : [_rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 67

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

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

✓ Solution by Mathematica

Time used: 30.591 (sec). Leaf size: 117

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

\[ \text {Solve}\left [\left \{x=-\frac {2 (2 K[1]-y(K[1]))}{K[1]^2},y(x)=\exp \left (\int _1^{K[1]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right ) \int _1^{K[1]}\frac {4 \exp \left (-\int _1^{K[3]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right )}{K[3]-2}dK[3]+c_1 \exp \left (\int _1^{K[1]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right )\right \},\{y(x),K[1]\}\right ] \]