60.1.115 problem 115

Internal problem ID [10129]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 115
Date solved : Tuesday, January 28, 2025 at 04:25:51 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x y^{\prime }-x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 50

dsolve(x*diff(y(x),x) - x*(y(x)-x)*sqrt(y(x)^2 + x^2) - y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 53.999 (sec). Leaf size: 154

DSolve[x*D[y[x],x] - x*(y[x]-x)*Sqrt[y[x]^2 + x^2] - y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x-8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )} \\ y(x)\to \frac {x+8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}