Internal problem ID [13333]
Internal file name [OUTPUT/12505_Wednesday_February_14_2024_11_54_26_PM_38663670/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.7 (j).
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}{{\mathrm e}^{-y}+1}d y &= x +c_{1}\\ y +\ln \left ({\mathrm e}^{-y}+1\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\ln \left ({\mathrm e}^{x +c_{1}}-1\right )\\ &=\ln \left (c_{1} {\mathrm e}^{x}-1\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left (c_{1} {\mathrm e}^{x}-1\right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left (c_{1} {\mathrm e}^{x}-1\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-{\mathrm e}^{-y}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{-y}+1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{{\mathrm e}^{-y}+1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{{\mathrm e}^{-y}+1}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left ({\mathrm e}^{-y}\right )+\ln \left ({\mathrm e}^{-y}+1\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left ({\mathrm e}^{x +c_{1}}-1\right ) \end {array} \]
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: 0.047 (sec). Leaf size: 11
dsolve(diff(y(x),x)=exp(-y(x))+1,y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (-1+c_{1} {\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 1.163 (sec). Leaf size: 32
DSolve[y'[x]==Exp[-y[x]]+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \log \left (-1+e^{x+c_1}\right ) \\ y(x)\to -i \pi \\ y(x)\to i \pi \\ \end{align*}