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)=

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. \[ \frac{\mathrm{dx}}{\mathrm{dt}}=7 e^{\mathrm{t}}-5 ; \quad \mathrm{x}(0)=5 \] \[ x(t)= \]
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Solution

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Solution Steps

Step 1: Identify the Differential Equation

We start with the given differential equation: dxdt=7et5 \frac{dx}{dt} = 7 e^{t} - 5 This equation describes the rate of change of x x with respect to t t .

Step 2: Integrate the Right-Hand Side

To find the general solution, we integrate the right-hand side: x(t)=(7et5)dt x(t) = \int (7 e^{t} - 5) \, dt Calculating the integral, we have: x(t)=7et5t+C x(t) = 7 e^{t} - 5t + C where C C is the constant of integration.

Step 3: Apply the Initial Condition

We are given the initial condition x(0)=5 x(0) = 5 . Substituting t=0 t = 0 into the equation: 5=7e05(0)+C 5 = 7 e^{0} - 5(0) + C This simplifies to: 5=7+C    C=57=2 5 = 7 + C \implies C = 5 - 7 = -2

Step 4: Write the Particular Solution

Substituting C C back into the general solution, we obtain the particular solution: x(t)=7et5t2 x(t) = 7 e^{t} - 5t - 2

Final Answer

The particular antiderivative that satisfies the given condition is: x(t)=7et5t2 \boxed{x(t) = 7 e^{t} - 5t - 2}

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