3.3.26 Laplace method

ode internal name "first_order_laplace"

These are ode’s solved using Laplace method. Currently only linear odes are supported. Both constant coefficients and time varying coefficients. For time varying only, only coefficients that are polynomial in \(t\) are allowed. For example the following ode can be solved using Laplace

\begin{align*} ty^{\prime }+y & =0\\ \left ( 1+t\right ) y^{\prime }+ty & =0\\ y^{\prime }+3t^{2}y & =0 \end{align*}

But not

\[ \sin \left ( t\right ) y^{\prime }+y=0 \]

Initial conditions can be at zero or not at zero or not given. For time varying, the ode is transform to Laplace using the property

\[\mathcal {L}\left ( t^{n}y\left ( t\right ) \right ) =\left ( -1\right ) ^{n}\frac {d^{n}}{ds^{n}}Y\left ( s\right ) \]

What this means, is that having \(t\) as coefficient will generate first order ode in \(Y\left ( s\right ) \) which needs to be solved first to find \(Y\left ( s\right ) \) before applying inverse Laplace transform to find the solution \(y\left ( t\right ) \). A coefficient \(t^{2}\) will generate second order ode in \(Y\left ( s\right ) \) and \(t^{3}\) will generate a third order ode in \(Y\left ( s\right ) \) and so on. This means if we are to use Laplace transform to solve first order ode, we could end having to solving an ode in \(Y\left ( s\right ) \) of much higher order and the generated solution \(Y\left ( s\right ) \) might become too complicated to even inverse Laplace it.

So it is not really useful to use Laplace method to solve time varying first order ode of coefficient of polynomial of power \(t^{n}\) where \(n>1\).

When the initial condition of the original ode is not at zero, the original condition must be shifted so it is at zero. This is more critical to do for time varying than for constant coefficients ode when we use Laplace transform method. This means we have to do change of variables first. See examples below.