Internal problem ID [11363]
Internal file name [OUTPUT/10346_Wednesday_May_17_2023_07_49_37_PM_13773109/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page
19
Problem number: 4(b).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x^{\prime }=\frac {1}{t \ln \left (t \right )}} \]
Integrating both sides gives \begin {align*} x &= \int { \frac {1}{t \ln \left (t \right )}\,\mathop {\mathrm {d}t}}\\ &= \ln \left (\ln \left (t \right )\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \ln \left (\ln \left (t \right )\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ x = \ln \left (\ln \left (t \right )\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }=\frac {1}{t \ln \left (t \right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int x^{\prime }d t =\int \frac {1}{t \ln \left (t \right )}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=\ln \left (\ln \left (t \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=\ln \left (\ln \left (t \right )\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 9
dsolve(diff(x(t),t)=1/(t*ln(t)),x(t), singsol=all)
\[ x \left (t \right ) = \ln \left (\ln \left (t \right )\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 11
DSolve[x'[t]==1/(t*Log[t]),x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \log (\log (t))+c_1 \]