Internal problem ID [8264]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 684.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y+x^{2} \sqrt {x^{2}+y^{2}}}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 4.131 (sec). Leaf size: 30
dsolve(diff(y(x),x) = (y(x)+(y(x)^2+x^2)^(1/2)*x^2)/x,y(x), singsol=all)
\[ \ln \left (\sqrt {x^{2}+y \relax (x )^{2}}+y \relax (x )\right )-\frac {x^{2}}{2}-\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 38.275 (sec). Leaf size: 70
DSolve[y'[x] == (y[x] + x^2*Sqrt[x^2 + y[x]^2])/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \tanh \left (\frac {x^2}{2}+c_1\right )}{\sqrt {\text {sech}^2\left (\frac {x^2}{2}+c_1\right )}} \\ y(x)\to \frac {x \tanh \left (\frac {x^2}{2}+c_1\right )}{\sqrt {\text {sech}^2\left (\frac {x^2}{2}+c_1\right )}} \\ \end{align*}