3 Consider the following initial value problem y ty 0 y0 1 I

3. Consider the following initial value problem y ty 0, y(0) 1 Is the method of Laplace transform applicable to this problem? If applicable, find the solution by the method. If not, explain why the Laplace method is not valid for this problem

Solution

This differential equation cannot be solved using laplace transform, as there is no way to find laplace transform of \"ty\" with y being unknown. That\'s why this differential equation cannot be solved using laplace transform.

It can be solved directly though.

y\'=ty

y\'/y = t

On integrating, we get:

ln(y) = t^2/2 + c

Using initial condiction, c=0

Therefore,

ln(y) = t^2/2

y = e^(t^2/2)