Internal problem ID [11969]
Internal file name [OUTPUT/10622_Saturday_September_02_2023_02_48_35_PM_21740883/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 5, Trivial differential equations. Exercises page 33
Problem number: 5.1 (iii).
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]
\[ \boxed {u^{\prime }=4 t \ln \left (t \right )} \]
Integrating both sides gives \begin {align*} u &= \int { 4 t \ln \left (t \right )\,\mathop {\mathrm {d}t}}\\ &= 2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} u &= 2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ u = 2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & u^{\prime }=4 t \ln \left (t \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & u^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int u^{\prime }d t =\int 4 t \ln \left (t \right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & u=2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \\ {} & {} & u=2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \end {array} \]
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: 18
dsolve(diff(u(t),t)=4*t*ln(t),u(t), singsol=all)
\[ u \left (t \right ) = 2 \ln \left (t \right ) t^{2}-t^{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 20
DSolve[u'[t]==4*t*Log[t],u[t],t,IncludeSingularSolutions -> True]
\[ u(t)\to -t^2+2 t^2 \log (t)+c_1 \]