75.11.1 problem 260

Internal problem ID [16852]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 260
Date solved : Tuesday, January 28, 2025 at 09:34:59 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} \left ({y^{\prime }}^{2}+1\right ) y^{2}-4 y^{\prime } y-4 x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.273 (sec). Leaf size: 71

dsolve((1+diff(y(x),x)^2)*y(x)^2-4*y(x)*diff(y(x),x)-4*x=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.472 (sec). Leaf size: 65

DSolve[(1+D[y[x],x]^2)*y[x]^2-4*y[x]*D[y[x],x]-4*x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {-4 x^2-4 (-4+c_1) x+16-c_1{}^2} \\ y(x)\to \frac {1}{2} \sqrt {-4 x^2-4 (-4+c_1) x+16-c_1{}^2} \\ \end{align*}