Internal problem ID [8919]
Internal file name [OUTPUT/7854_Monday_June_06_2022_12_47_21_AM_20934399/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5`[0, x*y*F(ln(ln(y))-ln(x))-y*ln(y)]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 136
dsolve(diff(y(x),x) = F(ln(ln(y(x)))-ln(x))*y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}\frac {F \left (\ln \left (\ln \left (y \left (x \right )\right )\right )-\ln \left (\textit {\_a} \right )\right )}{\textit {\_a} F \left (\ln \left (\ln \left (y \left (x \right )\right )\right )-\ln \left (\textit {\_a} \right )\right )-\ln \left (y \left (x \right )\right )}d \textit {\_a} -\left (\int _{}^{y \left (x \right )}\frac {1+\left (\int _{\textit {\_b}}^{x}\frac {-D\left (F \right )\left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )+F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )}{\left (\textit {\_a} F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )-\ln \left (\textit {\_f} \right )\right )^{2}}d \textit {\_a} \right ) \left (x F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (x \right )\right )-\ln \left (\textit {\_f} \right )\right )}{\textit {\_f} \left (x F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (x \right )\right )-\ln \left (\textit {\_f} \right )\right )}d \textit {\_f} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.22 (sec). Leaf size: 205
DSolve[y'[x] == F[-Log[x] + Log[Log[y[x]]]]*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{K[2] (x F(\log (\log (K[2]))-\log (x))-\log (K[2]))}-\int _1^x\left (\frac {F(\log (\log (K[2]))-\log (K[1])) \left (\frac {K[1] F'(\log (\log (K[2]))-\log (K[1]))}{K[2] \log (K[2])}-\frac {1}{K[2]}\right )}{(F(\log (\log (K[2]))-\log (K[1])) K[1]-\log (K[2]))^2}-\frac {F'(\log (\log (K[2]))-\log (K[1]))}{K[2] (F(\log (\log (K[2]))-\log (K[1])) K[1]-\log (K[2])) \log (K[2])}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F(\log (\log (y(x)))-\log (K[1]))}{F(\log (\log (y(x)))-\log (K[1])) K[1]-\log (y(x))}dK[1]=c_1,y(x)\right ] \]