Internal problem ID [12784]
Internal file name [OUTPUT/11437_Saturday_November_04_2023_08_47_19_AM_68303959/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }-y=0} \]
Since no initial condition is explicitly given, then let \begin {align*} y \left (0\right ) = c_{1} \end {align*}
Solving using the Laplace transform method. Let \[ \mathcal {L}\left (y\right ) =Y(s) \] Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (y^{\prime }\right )&= s Y(s) - y \left (0\right ) \end {align*}
The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s Y \left (s \right )-y \left (0\right )-Y \left (s \right ) = 0\tag {1} \end {align*}
Replacing initial condition gives \begin {align*} s Y \left (s \right )-c_{1} -Y \left (s \right ) = 0 \end {align*}
Solving for \(Y(s)\) gives \begin {align*} Y(s) = \frac {c_{1}}{s -1} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {c_{1}}{s -1}\right )\\ &= c_{1} {\mathrm e}^{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{x +c_{1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 4.609 (sec). Leaf size: 9
dsolve(diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{x} y \left (0\right ) \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 16
DSolve[y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^x \\ y(x)\to 0 \\ \end{align*}