Questions: Find the particular antiderivative of the following derivative that satisfies the given condition. dx/dt=7 e^t-5 ; x(0)=5 x(t)=

Transcript text: Find the particular antiderivative of the following derivative that satisfies the given condition. \[ \frac{\mathrm{dx}}{\mathrm{dt}}=7 e^{\mathrm{t}}-5 ; \quad \mathrm{x}(0)=5 \] \[ x(t)= \]

Solution

Solution Steps

Step 1: Identify the Differential Equation

We start with the given differential equation: \[ \frac{dx}{dt} = 7 e^{t} - 5 \] This equation describes the rate of change of \( x \) with respect to \( t \).

Step 2: Integrate the Right-Hand Side

To find the general solution, we integrate the right-hand side: \[ x(t) = \int (7 e^{t} - 5) \, dt \] Calculating the integral, we have: \[ x(t) = 7 e^{t} - 5t + C \] where \( C \) is the constant of integration.

Step 3: Apply the Initial Condition

We are given the initial condition \( x(0) = 5 \). Substituting \( t = 0 \) into the equation: \[ 5 = 7 e^{0} - 5(0) + C \] This simplifies to: \[ 5 = 7 + C \implies C = 5 - 7 = -2 \]

Step 4: Write the Particular Solution

Substituting \( C \) back into the general solution, we obtain the particular solution: \[ x(t) = 7 e^{t} - 5t - 2 \]