Internal problem ID [8263]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 683.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (-1+\ln \left (\left (x +1\right ) x \right ) y x^{4}-\ln \left (\left (x +1\right ) x \right ) x^{3}\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 155
dsolve(diff(y(x),x) = y(x)*(-1+ln(x*(x+1))*y(x)*x^4-ln(x*(x+1))*x^3)/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (x +1\right ) x \right )^{-\frac {x^{3}}{3}}}{\left (x +1\right )^{\frac {1}{3}} {\mathrm e}^{-\frac {2}{9} x^{3}+\frac {1}{6} x^{2}-\frac {1}{3} x} c_{1} x +x^{1-\frac {x^{3}}{3}} \left (x +1\right )^{-\frac {x^{3}}{3}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i x \left (x +1\right )\right )^{3} x^{3}}{6}-\frac {i \pi \mathrm {csgn}\left (i x \left (x +1\right )\right )^{2} \mathrm {csgn}\left (i x \right ) x^{3}}{6}-\frac {i \pi \mathrm {csgn}\left (i x \left (x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (x +1\right )\right ) x^{3}}{6}+\frac {i \pi \,\mathrm {csgn}\left (i x \left (x +1\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x +1\right )\right ) x^{3}}{6}}} \]
✓ Solution by Mathematica
Time used: 0.812 (sec). Leaf size: 54
DSolve[y'[x] == (y[x]*(-1 - x^3*Log[x*(1 + x)] + x^4*Log[x*(1 + x)]*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{x+c_1 e^{-\frac {1}{18} x (x (4 x-3)+6)} x \sqrt [3]{x+1} (x (x+1))^{\frac {x^3}{3}}} \\ y(x)\to 0 \\ \end{align*}