find the particular antiderivative of the following derivative that satisfies the given condition. dx/dt =…

Answer:

$7e^{t}-8t - 5$

Explanation:

Step1: Find general antiderivative

We know that the antiderivative of $e^{t}$ is $e^{t}$ and the antiderivative of a constant $a$ is $at$. So, integrating $\frac{dx}{dt}=7e^{t}-8$ with respect to $t$, we get $x(t)=7e^{t}-8t + C$, where $C$ is the constant of integration.

Step2: Use the initial - condition

Given $x(0) = 2$. Substitute $t = 0$ and $x(0)$ into $x(t)$: $x(0)=7e^{0}-8\times0 + C$. Since $e^{0}=1$, we have $2=7 + C$.

Step3: Solve for $C$

Subtract 7 from both sides of the equation $2 = 7+C$ to find $C$. So, $C=2 - 7=-5$.

Step4: Write the particular antiderivative

Substitute $C=-5$ into $x(t)=7e^{t}-8t + C$. We get $x(t)=7e^{t}-8t - 5$.