Internal problem ID [12616]
Internal file name [OUTPUT/11269_Friday_November_03_2023_06_29_32_AM_5344487/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.1, page 40
Problem number: 2 (C).
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}{1-y}d y &= x +c_{1}\\ -\ln \left (-1+y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-x -c_{1}}+1\\ &=\frac {{\mathrm e}^{-x}}{c_{1}}+1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-x}}{c_{1}}+1 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-x}}{c_{1}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{1-y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (1-y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-x -c_{1}}+1 \end {array} \]
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(x),x)=1-y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x}+1 \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 20
DSolve[y'[x]==1-y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+c_1 e^{-x} \\ y(x)\to 1 \\ \end{align*}