Internal problem ID [11977]
Internal file name [OUTPUT/10630_Saturday_September_02_2023_02_48_41_PM_84729893/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 7, Scalar autonomous ODEs. Exercises page 56
Problem number: 7.1 (i).
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]
\[ \boxed {x^{\prime }+x=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1-x}d x &= t +c_{1}\\ -\ln \left (x -1\right )&=t +c_{1} \end {align*}
Solving for \(x\) gives these solutions \begin {align*} x_1&={\mathrm e}^{-c_{1} -t}+1\\ &=\frac {{\mathrm e}^{-t}}{c_{1}}+1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {{\mathrm e}^{-t}}{c_{1}}+1 \\ \end{align*}
Verification of solutions
\[ x = \frac {{\mathrm e}^{-t}}{c_{1}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }+x=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=-x+1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{-x+1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{-x+1}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (-x+1\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=-{\mathrm e}^{-c_{1} -t}+1 \end {array} \]
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(x(t),t)=-x(t)+1,x(t), singsol=all)
\[ x \left (t \right ) = 1+{\mathrm e}^{-t} c_{1} \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 20
DSolve[x'[t]==-x[t]+1,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 1+c_1 e^{-t} \\ x(t)\to 1 \\ \end{align*}