Internal problem ID [9162]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 827.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {y^{\prime }+\frac {-y+x^{3} \sqrt {y^{2}+x^{2}}-y \sqrt {y^{2}+x^{2}}\, x^{2}}{x}=0} \]
✓ Solution by Maple
Time used: 0.078 (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)
\[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\frac {\sqrt {2}\, x^{3}}{3}-\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.329 (sec). Leaf size: 84
DSolve[y'[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 \tanh \left (\frac {x^3+3 c_1}{3 \sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {x^3+3 c_1}{3 \sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {x^3+3 c_1}{3 \sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}