Internal problem ID [8445]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 865.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\left (\frac {\ln \left (-1+y\right ) y}{\left (1-y\right ) \ln \relax (x ) x}-\frac {\ln \left (-1+y\right )}{\left (1-y\right ) \ln \relax (x ) x}-f \relax (x )\right ) \left (1-y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
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)
\[ y \relax (x ) = {\mathrm e}^{\left (\int \frac {f \relax (x )}{\ln \relax (x )}d x \right ) \ln \relax (x )} x^{c_{1}}+1 \]
✓ Solution by Mathematica
Time used: 0.253 (sec). Leaf size: 87
DSolve[y'[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]
\[ \text {Solve}\left [\int _1^x\left (-\frac {f(K[1])}{\log (K[1])}-\frac {\log (y(x)-1)}{K[1] \log ^2(K[1])}\right )dK[1]+\int _1^{y(x)}\left (\frac {1}{(K[2]-1) \log (x)}-\int _1^x-\frac {1}{K[1] (K[2]-1) \log ^2(K[1])}dK[1]\right )dK[2]=c_1,y(x)\right ] \]