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A second-order equation Consider the differential equation y”(t) – k2y(t) = 0, where k > 0 is a real number.
a. Verify by substitution that when k = 1, a solution of the equation is y(t) = C1et + C2e–t. You may assume that this function is the general solution.
b. Verify by substitution that when k = 2, the general solution of the equation is y(t) = C1e2t + C2e–2t.
c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.
d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C1 cosh kt + C2 sinh kt, where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively.
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