Question

find the particular antiderivative that satisfies the following conditions:
\\(\frac{dx}{dt} = 4e^t - 2\\); \\(x(0) = 3\\).

Explanation:

Step1: Integrate the derivative

To find the antiderivative of \(\frac{dx}{dt} = 4e^t - 2\), we integrate both sides with respect to \(t\). The integral of \(4e^t\) with respect to \(t\) is \(4e^t\) (since the integral of \(e^t\) is \(e^t\)) and the integral of \(-2\) with respect to \(t\) is \(-2t\). We also add the constant of integration \(C\). So,

$$ x(t)=\int(4e^t - 2)dt = 4e^t-2t + C $$

Step2: Use the initial condition to find \(C\)

We know that \(x(0) = 3\). Substitute \(t = 0\) and \(x(0)=3\) into the equation \(x(t)=4e^t-2t + C\):

$$ 3=4e^0-2(0)+C $$

Since \(e^0 = 1\), this simplifies to:

$$ 3 = 4(1)-0 + C $$

$$ 3=4 + C $$

Subtract 4 from both sides to solve for \(C\):

$$ C=3 - 4=-1 $$

Step3: Write the particular antiderivative

Substitute \(C=-1\) back into the equation for \(x(t)\):

$$ x(t)=4e^t-2t - 1 $$

Answer:

The particular antiderivative is \(x(t) = 4e^t - 2t - 1\)