Internal problem ID [535]
Internal file name [OUTPUT/535_Sunday_June_05_2022_01_43_49_AM_20194512/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 5.
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]
\[ \boxed {y^{\prime }-{\mathrm e}^{-y}=-1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-1+{\mathrm e}^{-y}}d y &= t +c_{1}\\ -y -\ln \left (-1+{\mathrm e}^{-y}\right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left ({\mathrm e}^{t +c_{1}}-1\right )-t -c_{1} \\ \end{align*}
Verification of solutions
\[ y = \ln \left ({\mathrm e}^{t +c_{1}}-1\right )-t -c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-{\mathrm e}^{-y}=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-1+{\mathrm e}^{-y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-1+{\mathrm e}^{-y}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{-1+{\mathrm e}^{-y}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (-1+{\mathrm e}^{-y}\right )+\ln \left ({\mathrm e}^{-y}\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left ({\mathrm e}^{t +c_{1}}-1\right )-t -c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 18
dsolve(diff(y(t),t) = -1+exp(-y(t)),y(t), singsol=all)
\[ y \left (t \right ) = -t +\ln \left ({\mathrm e}^{t +c_{1}}-1\right )-c_{1} \]
✓ Solution by Mathematica
Time used: 0.853 (sec). Leaf size: 21
DSolve[y'[t] == -1+Exp[-y[t]],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \log \left (1+e^{-t+c_1}\right ) \\ y(t)\to 0 \\ \end{align*}