60.2.107 problem 683

Internal problem ID [10694]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 683
Date solved : Monday, January 27, 2025 at 09:27:43 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {y \left (-1+\ln \left (x \left (1+x \right )\right ) y x^{4}-\ln \left (x \left (1+x \right )\right ) x^{3}\right )}{x} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 45

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)
 

\[ y = \frac {1}{x \left (\left (x \left (x +1\right )\right )^{\frac {x^{3}}{3}} c_{1} \left (x +1\right )^{{1}/{3}} {\mathrm e}^{-\frac {2}{9} x^{3}+\frac {1}{6} x^{2}-\frac {1}{3} x}+1\right )} \]

✓ Solution by Mathematica

Time used: 2.509 (sec). Leaf size: 232

DSolve[D[y[x],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 {e^{\frac {1}{18} x \left (4 x^2-3 x+6\right )} (x+1)^{2/3} (x (x+1))^{-\frac {x^3}{3}-1}}{-\int _1^x\frac {e^{\frac {1}{18} K[1] \left (4 K[1]^2-3 K[1]+6\right )} K[1]^2 (K[1] (K[1]+1))^{-\frac {1}{3} K[1]^3} \log (K[1] (K[1]+1))}{\sqrt [3]{K[1]+1}}dK[1]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {e^{\frac {1}{18} x \left (4 x^2-3 x+6\right )} (x+1)^{2/3} (x (x+1))^{-\frac {x^3}{3}-1}}{\int _1^x\frac {e^{\frac {1}{18} K[1] \left (4 K[1]^2-3 K[1]+6\right )} K[1]^2 (K[1] (K[1]+1))^{-\frac {1}{3} K[1]^3} \log (K[1] (K[1]+1))}{\sqrt [3]{K[1]+1}}dK[1]} \\ \end{align*}