60.1.387 problem 388

Internal problem ID [10401]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 388
Date solved : Monday, January 27, 2025 at 07:39:13 PM
CAS classification : [_dAlembert]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 223

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

\begin{align*} \frac {\frac {\left (y-\sqrt {y^{2}+2 x}\right ) \operatorname {arcsinh}\left (-y+\sqrt {y^{2}+2 x}\right )}{2}+x \sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}-2 c_{1} y+2 c_{1} \sqrt {y^{2}+2 x}}{\sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}} &= 0 \\ \frac {\frac {\left (-y-\sqrt {y^{2}+2 x}\right ) \operatorname {arcsinh}\left (y+\sqrt {y^{2}+2 x}\right )}{2}+x \sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}+2 c_{1} y+2 c_{1} \sqrt {y^{2}+2 x}}{\sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.405 (sec). Leaf size: 121

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

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