Please help me with this question.
2 answers:
You might be interested in
Answer:
The solution to the differential equation is
Step-by-step explanation:
Applying Laplace Transform will help us solve differential equations in Algebraic ways to find particular solutions, thus after applying Laplace transform and evaluating at the initial conditions we need to solve and apply Inverse Laplace transform to find the final answer.
Applying Laplace Transform
We can start applying Laplace at the given ODE
So we will get
Applying initial conditions and solving for X(s).
If we apply the initial conditions we get
Simplifying
Moving all terms that do not have X(s) to the other side
Factoring X(s) and moving the rest to the other side.
Partial fraction decomposition method.
In order to apply Inverse Laplace Transform, we need to separate the fractions into the simplest form, so we can apply partial fraction decomposition to the first 2 fractions. For the first one we have
So if we multiply both sides by the entire denominator we get
At this point we can find the value of A fast if we plug s = 0, so we get
So the value of A is
We can replace that on the previous equation and multiply all terms by 6
We can simplify a bit
And by comparing coefficients we can tell the values of B and C
So the separated fraction will be
We can repeat the process for the second fraction.
Multiplying by the entire denominator give us
We can plug the value of s = 1 to find A fast.
So we get
We can replace that on the previous equation and multiply all terms by 11
Simplifying
And by comparing coefficients we can tell the values of B and C.
So the separated fraction will be
So far replacing both expanded fractions on the solution
We can combine the fractions with the same denominator
Simplifying give us
Completing the square
One last step before applying the Inverse Laplace transform is to factor the denominators using completing the square procedure for this case, so we will have
We are adding half of the middle term but squared, so the first 3 terms become the perfect square, that is
So we get
Notice that the denominator has (s+2) inside a square we need to match that on the numerator so we can add and subtract 2
Lastly we can split the fraction one more
Applying Inverse Laplace Transform.
Since all terms are ready we can apply Inverse Laplace transform directly to each term and we will get