29.32.21 problem 955

Internal problem ID [5533]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 955
Date solved : Monday, January 27, 2025 at 11:35:44 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 2.056 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 4.176 (sec). Leaf size: 187

DSolve[(2 x -y[x]) (D[y[x],x])^2-2(1-x)D[y[x],x]+2-y[x]==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*}