5.4: Homework- Introduction to Differential Equations

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Describe as best you can at this point in your own words what a differential equation is.
Following earnings example from the previous chapter, if the number of employees in a company is growing at a rate of \(0.05\) times the number of employees, what is a differential equation that describes this situation?
\(E'(t) = 0.05 E(t)\).
ans
Verify the function \(f(x) = e^x - x - 1\) solves the differential equation:
\(f'(x) = f(x) + x\)

We see
\[\begin{align*} f'(x) & = f(x) + x \\ e^x - 1 & = (e^x - x - 1) + x \\ e^x - 1 & = e^x - 1 \end{align*}\]

as desired.

ans
Verify the function \(f(x) = 2 \sqrt{x}\) satisfies the differential equation:
\(f'(x) = \frac{2}{f(x)}.\)

We see

as desired.

ans
For each differential equation, find \(f'(t)\) for the given value of \(t\), or state there is not enough information.
1. Suppose \(f'(t) = 3 f(t) + 5\) and \(f(3) = -1\). Find \(f'(3)\).
  \(2\)
  ans
2. Suppose \(f'(t) = t + f(t)\), and \(f(7) = 1\). Find \(f'(7)\).
  \(8\)
  ans
3. Suppose \(f'(t) = \frac{1}{ \sqrt{f(t)} }\) and \(f(0) = 9\). Find \(f'(0)\).
  \(\frac{1}{3}\)
  ans
4. Suppose \(f'(t) = e^{-f(t)}\) and \(f(0) = 1\). Find \(f'(1)\).
  Not enough information.
  ans
For each relationship between the value of a function and its derivative, write down a differential equation. For example, if I said “a function is growing at a rate equal to seven times the value of the function” you’d write down \(f'(t) = 7 f(t)\).
1. A function is growing at a rate equal to twice the function value.
  \(f'(t) = 2 f(t)\)
  ans
2. A function is growing at a rate equal to the square root of the function value.
  \(f'(t) = \sqrt{f(t)}\)
  ans
3. A function is growing at a rate equal to \(t\) times the function value.
  \(f'(t) = t f(t)\)
  ans
4. A function is accelerating at a rate equal to the sum of the function value and how quickly the function is growing.
  \(f''(t) = f'(t) + f(t)\).
  ans
Verify that the given solution to each differential equation is correct.
1. Differential equation \(f'(t) = f(t) + 3\), solution \(f(t) = 3 e^t - 3\).
  
  \[\begin{align*} f'(t) & = f(t) + 3 \\ \frac{d}{dt}(3e^t - 3) & = (3e^t - 3) + 3 \\ 3e^t & = 3e^t \end{align*}\]
  
  ans
2. Differential equation \(f'(t) = 4\sqrt{f(t)}\), solution \(f(t) = 4 t^2\).
  
  \[\begin{align*} f'(t) & = 4 \sqrt{f(t)} \\ \frac{d}{dt}(4 t^2) & = 4 \sqrt{4 t^2} \\ 8t & = 4(2t) \\ 8t & = 8t. \end{align*}\]
  
  ans
3. Differential equation \(f'(t) = (f(t))^2\), solution \(f(t) = -t^{-1}\).
  
  \[\begin{align*} f'(t) & = (f(t))^2 \\ \frac{d}{dt} (-t^{-1}) & = (-t^{-1})^2 \\ t^{-2} & = t^{-2} \end{align*}\]
  
  ans
4. Differential equation \(f'(t) = e^{-f(t)}\), solution \(f(t) = \ln(t)\).
  
  \[\begin{align*} f'(t) & = e^{-f(t)} \\ \frac{d}{dt} \ln(t) & = e^{-\ln(t)} \\ \frac{1}{t} & = \frac{1}{e^{\ln(t)}} \\ \frac{1}{t} & = \frac{1}{t} \end{align*}\]
  
  ans