Internal problem ID [6369]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 78.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {\frac {y^{\prime } y}{1+\frac {\sqrt {1+\left (y^{\prime }\right )^{2}}}{2}}+x=0} \end {gather*}
✓ Solution by Maple
Time used: 2.42 (sec). Leaf size: 198
dsolve(diff(y(x),x)*y(x)/(1+1/2*sqrt(1+diff(y(x),x)^2))=-x,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\sqrt {-x^{2}+c_{1}}\, \left (1+\frac {\sqrt {\frac {x^{2}}{-x^{2}+c_{1}}+1}}{2}\right ) \\ y \relax (x ) = \sqrt {-x^{2}+c_{1}}\, \left (1+\frac {\sqrt {\frac {x^{2}}{-x^{2}+c_{1}}+1}}{2}\right ) \\ y \relax (x ) = -\frac {\sqrt {-9 x^{2}+15 c_{1}-6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \relax (x ) = \frac {\sqrt {-9 x^{2}+15 c_{1}-6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \relax (x ) = -\frac {\sqrt {-9 x^{2}+15 c_{1}+6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \relax (x ) = \frac {\sqrt {-9 x^{2}+15 c_{1}+6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.794 (sec). Leaf size: 153
DSolve[y'[x]*y[x]/(1+1/2*Sqrt[1+(y'[x])^2])==-x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (e^{c_1}-\sqrt {-9 x^2+4 e^{2 c_1}}\right ) \\ y(x)\to \frac {1}{3} \left (\sqrt {-9 x^2+4 e^{2 c_1}}+e^{c_1}\right ) \\ y(x)\to -\sqrt {-x^2+4 e^{2 c_1}}-e^{c_1} \\ y(x)\to \sqrt {-x^2+4 e^{2 c_1}}-e^{c_1} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}