60.2.9 problem 585

Internal problem ID [10596]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 585
Date solved : Tuesday, January 28, 2025 at 04:55:25 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.129 (sec). Leaf size: 138

dsolve(diff(y(x),x) = F(ln(ln(y(x)))-ln(x))*y(x),y(x), singsol=all)
 

\[ -\int _{\textit {\_b}}^{x}\frac {F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (\textit {\_a} \right )\right )}{\textit {\_a} F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (\textit {\_a} \right )\right )-\ln \left (y\right )}d \textit {\_a} -\int _{}^{y}\frac {-1+\left (x F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (x \right )\right )-\ln \left (\textit {\_f} \right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {D\left (F \right )\left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )-F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )}{\left (\textit {\_a} F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (\textit {\_a} \right )\right )-\ln \left (\textit {\_f} \right )\right )^{2}}d \textit {\_a} \right )}{\textit {\_f} \left (x F \left (\ln \left (\ln \left (\textit {\_f} \right )\right )-\ln \left (x \right )\right )-\ln \left (\textit {\_f} \right )\right )}d \textit {\_f} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 205

DSolve[D[y[x],x] == F[-Log[x] + Log[Log[y[x]]]]*y[x],y[x],x,IncludeSingularSolutions -> True]
 

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