Final answer:
To solve the given differential equation using power series, you assume the solution is represented as a power series and find a recurrence relation. The first 6 coefficients are computed using the recurrence relation.
Step-by-step explanation:
To solve the differential equation 2*y' - y = sinh(x) using power series, we can assume that the solution can be represented as a power series y = a0 + a1*x + a2*x^2 + a3*x^3 + ...
To find the recurrence relation, we substitute y into the equation and equate the coefficients of each power of x.
From the equation 2*y' - y = sinh(x), we can obtain the recurrence relation a_n = (a_{n-1} - (n+1)a_{n-2})/(2(n+1)), where a_0 = 0 and a_1 = 1.
Using this recurrence relation, we can compute the first 6 coefficients:
a0 = 0
a1 = 1
a2 = -1/6
a3 = -1/10
a4 = 1/144
a5 = 1/252