60.2.288 problem 865

Internal problem ID [10875]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 865
Date solved : Tuesday, January 28, 2025 at 05:27:42 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 19

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

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

✓ Solution by Mathematica

Time used: 0.349 (sec). Leaf size: 87

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

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