Finding x(t) with given coefficients can be a challenging task, especially when the problem involves working backwards. In this case, we are given the coefficients a_k and are asked to find x(t). It is important to note that the solution will depend on the value of k, as mentioned in the problem.
To start, we can use the given information to write the expression for x(t) as a sum of cosine and sine functions using Euler's formula:
x(t) = a_0 + ∑ (a_k*cos(k*t) + b_k*sin(k*t)) (1)
where a_k and b_k are the coefficients for the cosine and sine terms respectively. Since a_k = 0 if k=0, we can simplify the above equation as:
x(t) = a_0 + ∑ (b_k*sin(k*t)) (2)
Now, we can use the given expression for a_k and plug it into equation (2):
x(t) = a_0 + ∑ (j^k*sin(k*Pi/4)/(k*Pi)*sin(k*t)) (3)
Note that we have replaced the cosine term with its equivalent sine term using the identity cos(x) = sin(x+Pi/2).
Next, we can use the trigonometric identity sin(a)*sin(b) = 1/2*(cos(a-b) - cos(a+b)) to simplify equation (3):
x(t) = a_0 + 1/2∑ (j^k*(cos(k*(t-Pi/4)) - cos(k*(t+Pi/4)))/(k*Pi)) (4)
Finally, we can use the given period of 4 to simplify equation (4) further:
x(t) = a_0 + 1/2∑ (j^k*(cos(k*t) - cos(k*t+Pi)))/(k*Pi) (5)
Now, we can see that the solution for x(t) depends on the values of k and a_0. To find the specific value of x(t), we would need more information such as the initial conditions or the values of a_0 and k.
In conclusion, finding x(t) with given coefficients can be a challenging task, but by using the given information and using trigonometric identities, we can simplify the expression and write it in a more manageable form. However, the solution
