60.2.251 problem 827

Internal problem ID [10838]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 827
Date solved : Tuesday, January 28, 2025 at 05:22:29 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x} \end{align*}

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 50

dsolve(diff(y(x),x) = -(-y(x)+x^3*(y(x)^2+x^2)^(1/2)-x^2*(y(x)^2+x^2)^(1/2)*y(x))/x,y(x), singsol=all)
 

\[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 x^{2}+2 y^{2}}+y+x \right )}{y-x}\right )+\frac {\sqrt {2}\, x^{3}}{3}-\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 46.901 (sec). Leaf size: 172

DSolve[D[y[x],x] == (y[x] - x^3*Sqrt[x^2 + y[x]^2] + x^2*y[x]*Sqrt[x^2 + y[x]^2])/x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x-2 \sqrt {x^2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right ) \text {sech}^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )} \\ y(x)\to \frac {x+2 \sqrt {x^2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right ) \text {sech}^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {1}{3} \sqrt {2} \left (x^3+3 c_1\right )\right )} \\ y(x)\to x \\ \end{align*}