2.3.11 problem 11

Internal problem ID [8744]
Book : First order enumerated odes
Section : section 3. First order odes solved using Laplace method
Problem number : 11
Date solved : Tuesday, December 17, 2024 at 01:01:56 PM
CAS classification : [_separable]

Solve

\begin{align*} y^{\prime }+t^{2} y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=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*} t^{2} y &\xrightarrow {\mathscr {L}} \frac {d^{2}}{d s^{2}}Y \left (s \right )\\ y^{\prime } &\xrightarrow {\mathscr {L}} s Y \left (s \right )-y \left (0\right ) \end{align*}

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

\[ s Y-y \left (0\right )+Y^{\prime \prime } = 0 \]

Replacing \(y \left (0\right ) = 0\) in the above results in

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

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

This is Airy ODE. It has the general form

\[ a Y^{\prime \prime } + b Y^{\prime } + c s Y = F(s) \]

Where in this case

\begin{align*} a &= 1\\ b &= 0\\ c &= 1\\ F &= 0 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ Y = c_1 \operatorname {AiryAi}\left (-s \right )+c_2 \operatorname {AiryBi}\left (-s \right ) \]

Will add steps showing solving for IC soon.

Applying inverse Laplace transform on the above gives.

\begin{align*} y = c_1 \mathcal {L}^{-1}\left (\operatorname {AiryAi}\left (-s \right ), s , t\right )+c_2 \mathcal {L}^{-1}\left (\operatorname {AiryBi}\left (-s \right ), s , t\right )\tag {1} \end{align*}

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

\[ 0 = c_1 \mathcal {L}^{-1}\left (\operatorname {AiryAi}\left (-s \right ), s , t\right )+c_2 \mathcal {L}^{-1}\left (\operatorname {AiryBi}\left (-s \right ), s , t\right ) \]

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

\begin{align*} c_1 = -\frac {c_2 \mathcal {L}^{-1}\left (\operatorname {AiryBi}\left (-s \right ), s , t\right )}{\mathcal {L}^{-1}\left (\operatorname {AiryAi}\left (-s \right ), s , t\right )} \end{align*}

Substituting the above back into the solution (1) gives

\[ y = 0 \]

(a) Solution plot
\(y = 0\)

(b) Slope field plot
\(y^{\prime }+t^{2} y = 0\)

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }+y t^{2}=0, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y t^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-t^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y}d t =\int -t^{2}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\frac {t^{3}}{3}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\frac {t^{3}}{3}+\mathit {C1}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0={\mathrm e}^{\mathit {C1}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =\left (\right ) \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

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 : 0.071 (sec)
Leaf size : 40

dsolve([diff(y(t),t)+y(t)*t^2 = 0, 
        op([y(0) = 0])], 
        y(t),method=laplace)
 

\[ y = -\frac {c_2 \mathcal {L}^{-1}\left (\operatorname {AiryBi}\left (-\textit {\_s1} \right ), \textit {\_s1} , 0\right ) \mathcal {L}^{-1}\left (\operatorname {AiryAi}\left (-\textit {\_s1} \right ), \textit {\_s1} , t\right )}{\mathcal {L}^{-1}\left (\operatorname {AiryAi}\left (-\textit {\_s1} \right ), \textit {\_s1} , 0\right )}+c_2 \mathcal {L}^{-1}\left (\operatorname {AiryBi}\left (-\textit {\_s1} \right ), \textit {\_s1} , t\right ) \]

Mathematica DSolve solution

Solving time : 0.001 (sec)
Leaf size : 6

DSolve[{D[y[t],t]+t^2*y[t]==0,y[0]==0}, 
       y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 0 \]