Internal problem ID [2452]
Internal file name [OUTPUT/1944_Sunday_June_05_2022_02_40_17_AM_22616209/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-2, page 6
Problem number: 1.1-2 (f).
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]
\[ \boxed {y^{\prime }=\ln \left (t \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \ln \left (t \right )\,\mathop {\mathrm {d}t}}\\ &= t \ln \left (t \right )-t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= t \ln \left (t \right )-t +c_{1} \\ \end{align*}
Verification of solutions
\[ y = t \ln \left (t \right )-t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\ln \left (t \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int \ln \left (t \right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=t \ln \left (t \right )-t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=t \ln \left (t \right )-t +c_{1} \end {array} \]
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: 13
dsolve(diff(y(t),t)=ln(t),y(t), singsol=all)
\[ y \left (t \right ) = t \ln \left (t \right )-t +c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 15
DSolve[y'[t]==Log[t],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -t+t \log (t)+c_1 \]