Internal problem ID [2463]
Internal file name [OUTPUT/1955_Sunday_June_05_2022_02_40_48_AM_33782510/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-6, page 7
Problem number: 1.1-6 (a).
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]
\[ \boxed {y^{\prime }-y=-1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y -1}d y &= t +c_{1}\\ \ln \left (y -1\right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{t +c_{1}}+1\\ &={\mathrm e}^{t} c_{1} +1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{t} c_{1} +1 \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{t} c_{1} +1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y-1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y-1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y-1}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y-1\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{t +c_{1}}+1 \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 10
dsolve(diff(y(t),t)=y(t)-1,y(t), singsol=all)
\[ y \left (t \right ) = 1+{\mathrm e}^{t} c_{1} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 18
DSolve[y'[t]==y[t]-1,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to 1+c_1 e^t \\ y(t)\to 1 \\ \end{align*}