Internal problem ID [14421]
Internal file name [OUTPUT/14102_Monday_March_25_2024_09_49_32_PM_56596397/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Review exercises, page 80
Problem number: 20.
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]
\[ \boxed {y^{\prime }+y=5} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y +5}d y &= t +c_{1}\\ -\ln \left (y -5\right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-t -c_{1}}+5\\ &=\frac {{\mathrm e}^{-t}}{c_{1}}+5 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-t}}{c_{1}}+5 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-t}}{c_{1}}+5 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y=5 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y+5 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y+5}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{-y+5}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (-y+5\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-t -c_{1}}+5 \end {array} \]
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)+y(t)=5,y(t), singsol=all)
\[ y \left (t \right ) = 5+{\mathrm e}^{-t} c_{1} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 20
DSolve[y'[t]+y[t]==5,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to 5+c_1 e^{-t} \\ y(t)\to 5 \\ \end{align*}