Ok I know how to do this but what do I need to do?
The Laplace of the given equation is
.
According to the statement
we have given that the equation and we have to solve this problem with the help of the Laplace transform.
So, According to the statement
the given equation is
y'' − 4y' + 4y = t^3e^2t, y(0) = 0, y'(0) = 0
And the Laplace transform is an integral transform method which is particularly useful in solving linear ordinary differential equations.
Firstly fill the value of the Laplace of the second order derivative and put x = 0 in the equation
And same it with the first order of the derivative.
And then the Laplace of the equation become

So, The Laplace of the given equation is 
Learn more about Laplace transform here
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The solution to the Questions are
- The alternate hypothesis demonstrates that there are two possible outcomes for the test.
- Decision rule: If z > 2.05 or z<-2.05, reject H0
- The two-tailed nature of the test is shown by the alternative hypothesis.
- The result of the test yields the following P-value: 0.0096
<h3>What is the alternate hypothesis?</h3>
(a)
The alternate hypothesis demonstrates that there are two possible outcomes for the test.
(b)
Here we have

(b)
Here the test is two-tailed. So for
, the critical values of the z-test are -2.05 and 2.05.
Decision rule: If z > 2.05 or z<-2.05, reject H0
(c)
Test statistics will be

z=2.59
(d)
The two-tailed nature of the test is shown by the alternative hypothesis.
(e)
The result of the test yields the following P-value: 0.0096
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The square root of 32 is 5.65685