56.1.78 problem 78

Internal problem ID [8790]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 78
Date solved : Monday, January 27, 2025 at 04:58:45 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}}&=-x \end{align*}

✓ Solution by Maple

Time used: 1.737 (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 &= -\frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\ y &= \frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\ y &= -\frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y &= \frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y &= -\frac {\sqrt {-9 x^{2}+15 c_{1} +6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\ y &= \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: 2.152 (sec). Leaf size: 153

DSolve[D[y[x],x]*y[x]/(1+1/2*Sqrt[1+(D[y[x],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*}