Internal problem ID [2624]
Internal file name [OUTPUT/2116_Sunday_June_05_2022_02_49_06_AM_88882434/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential
Equations. page 43
Problem number: Problem 3.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime } {\mathrm e}^{y+x}=1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= {\mathrm e}^{-x} {\mathrm e}^{-y} \end {align*}
Where \(f(x)={\mathrm e}^{-x}\) and \(g(y)={\mathrm e}^{-y}\). Integrating both sides gives \begin{align*} \frac {1}{{\mathrm e}^{-y}} \,dy &= {\mathrm e}^{-x} \,d x \\ \int { \frac {1}{{\mathrm e}^{-y}} \,dy} &= \int {{\mathrm e}^{-x} \,d x} \\ {\mathrm e}^{y}&=-{\mathrm e}^{-x}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= -x +\ln \left ({\mathrm e}^{x} c_{1} -1\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -x +\ln \left ({\mathrm e}^{x} c_{1} -1\right ) \\ \end{align*}
Verification of solutions
\[ y = -x +\ln \left ({\mathrm e}^{x} c_{1} -1\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } {\mathrm e}^{y+x}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{{\mathrm e}^{y+x}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } {\mathrm e}^{y}=\frac {1}{{\mathrm e}^{x}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } {\mathrm e}^{y}d x =\int \frac {1}{{\mathrm e}^{x}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & {\mathrm e}^{y}=-\frac {1}{{\mathrm e}^{x}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x +\ln \left ({\mathrm e}^{x} c_{1} -1\right ) \end {array} \]
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: 0.0 (sec). Leaf size: 15
dsolve(exp(x+y(x))*diff(y(x),x)-1=0,y(x), singsol=all)
\[ y \left (x \right ) = \ln \left ({\mathrm e}^{x} c_{1} -1\right )-x \]
✓ Solution by Mathematica
Time used: 0.097 (sec). Leaf size: 16
DSolve[Exp[x+y[x]]*y'[x]-1==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \log \left (-e^{-x}+c_1\right ) \]