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:
dtdx=7et−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)=∫(7et−5)dt
Calculating the integral, we have:
x(t)=7et−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=7e0−5(0)+C
This simplifies to:
5=7+C⟹C=5−7=−2
Step 4: Write the Particular Solution
Substituting C back into the general solution, we obtain the particular solution:
x(t)=7et−5t−2
Final Answer
The particular antiderivative that satisfies the given condition is:
x(t)=7et−5t−2