29.7.22 problem 197

Internal problem ID [4797]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 197
Date solved : Tuesday, January 28, 2025 at 02:39:48 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 50

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

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

✓ Solution by Mathematica

Time used: 53.906 (sec). Leaf size: 154

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