Internal problem ID [3978]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 732.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y^{2} x +\left (x^{2}-y^{2}\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(x*y(x)*(x+sqrt(x^2-y(x)^2))*diff(y(x),x) = x*y(x)^2-(x^2-y(x)^2)^(3/2),y(x), singsol=all)
\[ -\frac {2 \sqrt {x^{2}-y \left (x \right )^{2}}}{x}+\frac {y \left (x \right )^{2}}{x^{2}}+2 \ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 26.912 (sec). Leaf size: 385
DSolve[x y[x](x+Sqrt[x^2-y[x]^2])y'[x]==x y[x]^2-(x^2-y[x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} y(x)\to \sqrt {-2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} y(x)\to -\sqrt {2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} y(x)\to \sqrt {2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} y(x)\to -\sqrt {-2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} y(x)\to \sqrt {-2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} y(x)\to -\sqrt {2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} y(x)\to \sqrt {2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} \end{align*}