Find f if f′′(t)=2e^t+3sin(t), f(0)=−4, f(π)=−5.
To find the function f, we need to integrate f''(t) twice with respect to t, and then use the given initial conditions f(0) = -4 and f(π) = -5 to determine the values of the integration constants.
Step 1: Integrate f''(t) with respect to t to find f'(t):
∫ f''(t) dt = ∫ (2e^t + 3sin(t)) dt
Using the properties of integration, we can split the integral into two separate integrals:
∫ 2e^t dt + ∫ 3sin(t) dt
Integrating each term separately:
∫2e^t dt = 2∫e^t dt = 2e^t + C1, where C1 is the first integration constant.
∫3sin(t) dt = -3cos(t) + C2, where C2 is the second integration constant.
Therefore, f'(t) = 2e^t - 3cos(t) + C1 + C2.
Step 2: Integrate f'(t) with respect to t to find f(t):
∫ f'(t) dt = ∫ (2e^t - 3cos(t) + C1 + C2) dt
Again, we can integrate each term separately:
∫ 2e^t dt - ∫ 3cos(t) dt + ∫ C1 dt + ∫ C2 dt
∫ 2e^t dt = 2∫ e^t dt = 2e^t + C3, where C3 is the third integration constant.
∫ 3cos(t) dt = 3∫ cos(t) dt = 3sin(t) + C4, where C4 is the fourth integration constant.
∫ C1 dt = C1t + C5, where C5 is the fifth integration constant.
∫ C2 dt = C2t + C6, where C6 is the sixth integration constant.
Combining all the terms, we get:
f(t) = 2e^t - 3sin(t) + C1t + C2 + C3t + C4 + C5 + C6
Simplifying the constants, we can write:
f(t) = (2e^t - 3sin(t))t + (C1t + C2) + (C3t + C4 + C5 + C6)
Step 3: Use the initial conditions to find the values of the integration constants:
Given f(0) = -4, we can substitute t = 0 into the function f(t) and solve for the constants:
f(0) = (2e^0 - 3sin(0))0 + (C1(0) + C2) + (C3(0) + C4 + C5 + C6) = C2 + C4 + C5 + C6 = -4
Given f(π) = -5, we can substitute t = π into the function f(t) and solve for the constants:
f(π) = (2e^π - 3sin(π))π + (C1π + C2) + (C3π + C4 + C5 + C6) = -5
We now have two equations:
C2 + C4 + C5 + C6 = -4
(2e^π - 3sin(π))π + (C1π + C2) + (C3π + C4 + C5 + C6) = -5
Solve these equations to find the values of the constants C2, C4, C5, and C6, and substitute them back into the function f(t).
f' = 2e^t - 3cost + C
f(0) = 2-3+C = -4 so C = -3
that makes f'(t) = 2e^t - 3cost - 3
Now do that again for f(t)