Internal problem ID [8220]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 640.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y}{\ln \left (\ln \relax (y)\right )-\ln \relax (x )+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.896 (sec). Leaf size: 126
dsolve(diff(y(x),x) = 1/(ln(ln(y(x)))-ln(x)+1)*y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}-\frac {1}{\ln \left (y \relax (x )\right ) \ln \left (\textit {\_a} \right )-\ln \left (y \relax (x )\right ) \ln \left (\ln \left (y \relax (x )\right )\right )-\ln \left (y \relax (x )\right )+\textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {-\ln \left (\ln \left (\textit {\_f} \right )\right )+\ln \relax (x )-1}{\left (\ln \left (\textit {\_f} \right ) \ln \relax (x )-\ln \left (\textit {\_f} \right ) \ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_f} \right )+x \right ) \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\frac {\frac {\ln \left (\textit {\_a} \right )}{\textit {\_f}}-\frac {\ln \left (\ln \left (\textit {\_f} \right )\right )}{\textit {\_f}}-\frac {2}{\textit {\_f}}}{\left (\ln \left (\textit {\_f} \right ) \ln \left (\textit {\_a} \right )-\ln \left (\textit {\_f} \right ) \ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_f} \right )+\textit {\_a} \right )^{2}}d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 53
DSolve[y'[x] == y[x]/(1 - Log[x] + Log[Log[y[x]]]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {\log (x)-\log (\log (K[1]))-1}{K[1] (x+\log (x) \log (K[1])-\log (K[1])-\log (K[1]) \log (\log (K[1])))}dK[1]=c_1,y(x)\right ] \]