60.2.186 problem 762

Internal problem ID [10773]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 762
Date solved : Tuesday, January 28, 2025 at 05:13:27 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 22

dsolve(diff(y(x),x) = -(ln(y(x))*x+ln(y(x))-x)*y(x)/x/(x+1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 60

DSolve[D[y[x],x] == ((x - Log[y[x]] - x*Log[y[x]])*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {x}{K[2]}-\int _1^x-\frac {1}{K[2]}dK[1]\right )dK[2]+\int _1^x\left (-\log (y(x))-\frac {1}{K[1]+1}+1\right )dK[1]=c_1,y(x)\right ] \]