29.26.1 problem 732

Internal problem ID [5322]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 732
Date solved : Monday, January 27, 2025 at 11:12:13 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime }&=x y^{2}-\left (x^{2}-y^{2}\right )^{{3}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 40

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)
 

\[ \frac {2 \ln \left (x \right ) x^{2}-c_{1} x^{2}+y \left (x \right )^{2}-2 x \sqrt {x^{2}-y \left (x \right )^{2}}}{x^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 22.831 (sec). Leaf size: 385

DSolve[x*y[x]*(x+Sqrt[x^2-y[x]^2])*D[y[x],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*}