find the particular antiderivative of the following derivative that satisfies the given condition.\n\\(…

find the particular antiderivative of the following derivative that satisfies the given condition.\n\\( \\frac { d x } { d t } = 8 e ^ { t } - 5 ; \\quad x ( 0 ) = 2 \\)\n\\( x ( t ) = \\square \\)

find the particular antiderivative of the following derivative that satisfies the given condition.\n\\( \\frac { d x } { d t } = 8 e ^ { t } - 5 ; \\quad x ( 0 ) = 2 \\)\n\\( x ( t ) = \\square \\)

Answer

Explanation:

Step1: Integrate the derivative

We know that if (\frac{dx}{dt}=8e^{t}-5), then by the fundamental theorem of calculus (x(t)=\int(8e^{t}-5)dt). Using the integral rules (\int e^{t}dt = e^{t}+C_1) and (\int kdt=kt + C_2) ((k) is a constant), we have: (x(t)=8\int e^{t}dt-5\int dt) (x(t)=8e^{t}-5t + C)

Step2: Use the initial condition

Given (x(0) = 2). Substitute (t = 0) and (x=2) into (x(t)=8e^{t}-5t + C): (2=8e^{0}-5\times0 + C) Since (e^{0}=1), we get (2=8\times1-0 + C) (C=2 - 8=-6)

Answer:

(x(t)=8e^{t}-5t - 6)