# 5.4: Homework- Introduction to Differential Equations

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1. Describe as best you can at this point in your own words what a differential equation is.
2. 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
3. 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
4. Verify the function $$f(x) = 2 \sqrt{x}$$ satisfies the differential equation:

$$f'(x) = \frac{2}{f(x)}.$$

We see

as desired.

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5. 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 $$f'(3)$$.
$$2$$
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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.
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6. 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.
$$f'(t) = \sqrt{f(t)}$$
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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.
$$f''(t) = f'(t) + f(t)$$.
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7. 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*}

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