Internal problem ID [14978]
Internal file name [OUTPUT/14988_Monday_April_15_2024_12_04_46_AM_7008515/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 4. Equations with variables separable and equations reducible to them. Exercises
page 38
Problem number: 52.
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]
\[ \boxed {{\mathrm e}^{-y} y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int {\mathrm e}^{-y}d y &= x +c_{1}\\ -{\mathrm e}^{-y}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\ln \left (-x -c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (-x -c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = -\ln \left (-x -c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-y} y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int {\mathrm e}^{-y} y^{\prime }d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -{\mathrm e}^{-y}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (-x -c_{1} \right ) \end {array} \]
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: 12
dsolve(exp(-y(x))*diff(y(x),x)=1,y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (-\frac {1}{c_{1} +x}\right ) \]
✓ Solution by Mathematica
Time used: 0.07 (sec). Leaf size: 16
DSolve[Exp[-y[x]]*y'[x]==1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\log (-x-c_1) \]