60.2.187 problem 763

Internal problem ID [10774]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 763
Date solved : Monday, January 27, 2025 at 09:43:30 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: 14

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

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

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 72

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

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