2.3.4 problem 4

Internal problem ID [8737]
Book : First order enumerated odes
Section : section 3. First order odes solved using Laplace method
Problem number : 4
Date solved : Tuesday, December 17, 2024 at 01:01:51 PM
CAS classification : [_separable]

Solve

\begin{align*} t y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \end{align*}

We will now apply Laplace transform to each term in the ode. Since this is time varying, the following Laplace transform property will be used

\begin{align*} t^{n} f \left (t \right ) &\xrightarrow {\mathscr {L}} (-1)^n \frac {d^n}{ds^n} F(s) \end{align*}

Where in the above \(F(s)\) is the laplace transform of \(f \left (t \right )\). Applying the above property to each term of the ode gives

\begin{align*} y &\xrightarrow {\mathscr {L}} Y \left (s \right )\\ t y^{\prime } &\xrightarrow {\mathscr {L}} -Y \left (s \right )-s \left (\frac {d}{d s}Y \left (s \right )\right ) \end{align*}

Collecting all the terms above, the ode in Laplace domain becomes

\[ -s Y^{\prime } = 0 \]

The above ode in Y(s) is now solved.

Since the ode has the form \(Y^{\prime }=f(s)\), then we only need to integrate \(f(s)\).

\begin{align*} \int {dY} &= \int {0\, ds} + c_1 \\ Y &= c_1 \end{align*}

Applying inverse Laplace transform on the above gives.

\begin{align*} y = c_1 \delta \left (t \right )\tag {1} \end{align*}

Substituting initial conditions \(y \left (0\right ) = y_{0}\) and \(y^{\prime }\left (0\right ) = y_{0}\) into the above solution Gives

\[ y_{0} = c_1 \delta \left (0\right ) \]

Solving for the constant \(c_1\) from the above equation gives

\begin{align*} c_1 = \frac {y_{0}}{\delta \left (0\right )} \end{align*}

Substituting the above back into the solution (1) gives

\[ y = \frac {y_{0} \delta \left (t \right )}{\delta \left (0\right )} \]

Figure 2.101: Slope field plot
\(t y^{\prime }+y = 0\)

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [t y^{\prime }+y=0, y \left (0\right )=y_{0} \right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {1}{t} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y}d t =\int -\frac {1}{t}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (t \right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{\mathit {C1}}}{t} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

Maple dsolve solution

Solving time : 0.020 (sec)
Leaf size : 12

dsolve([t*diff(y(t),t)+y(t) = 0, 
        op([y(0) = y__0])], 
        y(t),method=laplace)
 

\[ y = \frac {y_{0} \delta \left (t \right )}{\delta \left (0\right )} \]

Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{t*D[y[t],t]+y[t]==0,y[0]==y0}, 
       y[t],t,IncludeSingularSolutions->True]
 

Not solved