Internal problem ID [7122]
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]
\[ \boxed {\frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}}=-x} \]
✓ Solution by Maple
Time used: 1.891 (sec). Leaf size: 187
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 \left (x \right ) &= -\frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\ y \left (x \right ) &= \frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \left (x \right ) &= -\frac {\sqrt {-9 x^{2}+15 c_{1} +6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y \left (x \right ) &= \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: 2.255 (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*}