Find all functions $f(x)$ with the following properties. \[ f^{\prime}(x)=0.6 e^{-0.7 x}, f(0)=5 \] \[ f(x)=\square \]
Solution
Step 1 ：The problem is asking for a function \(f(x)\) that satisfies the given differential equation and initial condition. The differential equation is a first order ordinary differential equation (ODE) and can be solved by integrating both sides with respect to \(x\). The constant of integration can be determined by using the initial condition \(f(0)=5\).
Step 2 ：Integrate the differential equation \(f'(x) = 0.6e^{-0.7x}\) with respect to \(x\) to get \(f(x) = C1 - 0.857142857142857e^{-0.7x}\).
Step 3 ：Substitute the initial condition \(f(0) = 5\) into the equation to solve for \(C1\). This gives \(C1 = 5.85714285714286\).
Step 4 ：Substitute \(C1\) back into the equation to get the final solution \(f(x) = 5.85714285714286 - 0.857142857142857e^{-0.7x}\).
Step 5 ：Final Answer: The function \(f(x)\) that satisfies the given properties is \(\boxed{5.85714285714286 - 0.857142857142857e^{-0.7x}}\).
