Internal problem ID [11370]
Internal file name [OUTPUT/10353_Wednesday_May_17_2023_07_49_48_PM_88378866/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 1.3.1 Separable equations. Exercises
page 26
Problem number: 1(d).
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]
\[ \boxed {u^{\prime }-\frac {1}{5-2 u}=0} \]
Integrating both sides gives \begin {align*} \int \left (5-2 u \right )d u &= t +c_{1}\\ -u \left (u -5\right )&=t +c_{1} \end {align*}
Solving for \(u\) gives these solutions \begin {align*} u_1&=\frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2}\\ u_2&=\frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} u &= \frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ \tag{2} u &= \frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ \end{align*}
Verification of solutions
\[ u = \frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2} \] Verified OK.
\[ u = \frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & u^{\prime }-\frac {1}{5-2 u}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & u^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }=\frac {1}{5-2 u} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \left (5-2 u\right ) u^{\prime }=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (5-2 u\right ) u^{\prime }d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -u^{2}+5 u=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \\ {} & {} & \left \{u=\frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2}, u=\frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(diff(u(t),t)=1/(5-2*u(t)),u(t), singsol=all)
\begin{align*} u \left (t \right ) &= \frac {5}{2}-\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ u \left (t \right ) &= \frac {5}{2}+\frac {\sqrt {25-4 t -4 c_{1}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.142 (sec). Leaf size: 49
DSolve[u'[t]==1/(5-2*u[t]),u[t],t,IncludeSingularSolutions -> True]
\begin{align*} u(t)\to \frac {1}{2} \left (5-\sqrt {-4 t+25+4 c_1}\right ) \\ u(t)\to \frac {1}{2} \left (5+\sqrt {-4 t+25+4 c_1}\right ) \\ \end{align*}