Internal problem ID [2464]
Internal file name [OUTPUT/1956_Sunday_June_05_2022_02_40_50_AM_17490583/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-6, page 7
Problem number: 1.1-6 (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]
\[ \boxed {y^{\prime }+y=1} \]
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\\ &=\frac {{\mathrm e}^{-t}}{c_{1}}+1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-t}}{c_{1}}+1 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-t}}{c_{1}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{1-y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (1-y\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-t -c_{1}}+1 \end {array} \]
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: 12
dsolve(diff(y(t),t)=1-y(t),y(t), singsol=all)
\[ y \left (t \right ) = 1+{\mathrm e}^{-t} c_{1} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 20
DSolve[y'[t]==1-y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to 1+c_1 e^{-t} \\ y(t)\to 1 \\ \end{align*}