Internal problem ID [8163]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 585.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {y^{\prime }-F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 163
dsolve(diff(y(x),x) = F(ln(ln(y(x)))-ln(x))*y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}\frac {F \left (\ln \left (\ln \left (y \left (x \right )\right )\right )-\ln \left (\textit {\_a} \right )\right )}{-\textit {\_a} F \left (\ln \left (\ln \left (y \left (x \right )\right )\right )-\ln \left (\textit {\_a} \right )\right )+\ln \left (y \left (x \right )\right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {1}{\textit {\_f} \left (-x F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (x \right )\right )+\ln \left (\textit {\_f} \right )\right )}-\left (\int _{\textit {\_b}}^{x}\left (\frac {D\left (F \right )\left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )}{\textit {\_f} \ln \left (\textit {\_f} \right ) \left (-\textit {\_a} F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )+\ln \left (\textit {\_f} \right )\right )}-\frac {F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right ) \left (-\frac {\textit {\_a} D\left (F \right )\left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )}{\textit {\_f} \ln \left (\textit {\_f} \right )}+\frac {1}{\textit {\_f}}\right )}{\left (-\textit {\_a} F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )+\ln \left (\textit {\_f} \right )\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.217 (sec). Leaf size: 205
DSolve[y'[x] == F[-Log[x] + Log[Log[y[x]]]]*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{K[2] (x F(\log (\log (K[2]))-\log (x))-\log (K[2]))}-\int _1^x\left (\frac {F(\log (\log (K[2]))-\log (K[1])) \left (\frac {K[1] F'(\log (\log (K[2]))-\log (K[1]))}{K[2] \log (K[2])}-\frac {1}{K[2]}\right )}{(F(\log (\log (K[2]))-\log (K[1])) K[1]-\log (K[2]))^2}-\frac {F'(\log (\log (K[2]))-\log (K[1]))}{K[2] (F(\log (\log (K[2]))-\log (K[1])) K[1]-\log (K[2])) \log (K[2])}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F(\log (\log (y(x)))-\log (K[1]))}{F(\log (\log (y(x)))-\log (K[1])) K[1]-\log (y(x))}dK[1]=c_1,y(x)\right ] \]