60.2.135 problem 711

Internal problem ID [10722]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 711
Date solved : Tuesday, January 28, 2025 at 05:08:37 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 )-1\right ) y}{1+x} \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 27

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

\[ y = {\mathrm e}^{c_{1} {\mathrm e}^{-x}-\operatorname {Ei}_{1}\left (-x -1\right ) {\mathrm e}^{-x -1}} \]

✓ Solution by Mathematica

Time used: 0.261 (sec). Leaf size: 70

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

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