60.1.470 problem 473

Internal problem ID [10484]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 473
Date solved : Monday, January 27, 2025 at 07:55:10 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 1.689 (sec). Leaf size: 71

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

\begin{align*} y &= -\sqrt {2}\, x +\sqrt {2}+x +1 \\ y &= \left (x -1\right ) \sqrt {2}+x +1 \\ y &= 2+\frac {c_{1}}{2}-\frac {\sqrt {c_{1} \left (-c_{1} +4 x -4\right )}}{2} \\ y &= 2+c_{1} -\sqrt {c_{1} \left (-c_{1} +2 x -2\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.209 (sec). Leaf size: 187

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {-e^{c_1} \left (4 x-4+e^{c_1}\right )}+2-\frac {e^{c_1}}{2} \\ y(x)\to \frac {1}{2} \left (\sqrt {-e^{c_1} \left (4 x-4+e^{c_1}\right )}+4-e^{c_1}\right ) \\ y(x)\to -\sqrt {-e^{c_1} \left (2 x-2+e^{c_1}\right )}+2-e^{c_1} \\ y(x)\to \sqrt {-e^{c_1} \left (2 x-2+e^{c_1}\right )}+2-e^{c_1} \\ y(x)\to 2 \\ y(x)\to x-\sqrt {2} \sqrt {(x-1)^2}+1 \\ y(x)\to x+\sqrt {2} \sqrt {(x-1)^2}+1 \\ \end{align*}