Internal problem ID [534]
Internal file name [OUTPUT/534_Sunday_June_05_2022_01_43_47_AM_54468129/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 4.
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*} y_1&=\ln \left (-\frac {1}{{\mathrm e}^{t +c_{1}}-1}\right )\\ &=\ln \left (-\frac {1}{c_{1} {\mathrm e}^{t}-1}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (-\frac {1}{c_{1} {\mathrm e}^{t}-1}\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (-\frac {1}{c_{1} {\mathrm e}^{t}-1}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-{\mathrm e}^{y}=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-1+{\mathrm e}^{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-1+{\mathrm e}^{y}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{-1+{\mathrm e}^{y}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (-1+{\mathrm e}^{y}\right )-\ln \left ({\mathrm e}^{y}\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (-\frac {1}{{\mathrm e}^{t +c_{1}}-1}\right ) \end {array} \]
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: 0.0 (sec). Leaf size: 15
dsolve(diff(y(t),t) = -1+exp(y(t)),y(t), singsol=all)
\[ y \left (t \right ) = \ln \left (-\frac {1}{{\mathrm e}^{t} c_{1} -1}\right ) \]
✓ Solution by Mathematica
Time used: 0.817 (sec). Leaf size: 28
DSolve[y'[t]== -1+Exp[y[t]],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \log \left (\frac {1}{2} \left (1-\tanh \left (\frac {t+c_1}{2}\right )\right )\right ) \\ y(t)\to 0 \\ \end{align*}