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)=

Solution Steps

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 $t$.

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.

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: $$5 = 7 + C \implies C = 5 - 7 = -2$$

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

Final Answer