We are given the differential equation:
with initial conditions .
To find the Laplace transform of , we can apply the Laplace transform to both sides of the differential equation:
Using the property of the Laplace transform that and , we can rewrite the equation as:
Since and , we can simplify the equation to:
Simplifying further, we get:
Therefore, the Laplace transform of is:
To solve for , we can use partial fraction decomposition. First, we factor the denominator:
Thus, we can write:
Simplifying, we get:
Using the initial condition , we know that . Thus, we can solve for :
Using the initial condition , we know that . Thus, we can solve for :
Now we can write in terms of and :
To solve for and , we can take the Laplace inverse of using tables or the method of partial fraction decomposition:
where and are constants that we still need to solve for.
Using the initial condition , we can solve for :
Thus, the solution to the differential equation with initial conditions is:
Using the initial condition , we get:
Therefore, the solution to the differential equation with initial conditions is .
In terms of Laplace transform, we have:
where and .