Internal problem ID [5734]
Internal file name [OUTPUT/4982_Sunday_June_05_2022_03_15_53_PM_52181844/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 21.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-{\mathrm e}^{x -y}=0} \]
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 &= \ln \left ({\mathrm e}^{x}+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \ln \left ({\mathrm e}^{x}+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \ln \left ({\mathrm e}^{x}+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-{\mathrm e}^{x -y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{x -y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } {\mathrm e}^{y}={\mathrm e}^{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } {\mathrm e}^{y}d x =\int {\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & {\mathrm e}^{y}={\mathrm e}^{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left ({\mathrm e}^{x}+c_{1} \right ) \end {array} \]
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: 9
dsolve(diff(y(x),x)=exp(x-y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \ln \left ({\mathrm e}^{x}+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.743 (sec). Leaf size: 12
DSolve[y'[x]==Exp[x-y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \log \left (e^x+c_1\right ) \]