1.1 Some Basic Mathematical Models: Direction Field

differential equation: an equation that contains derivatives

for example,

\[ \frac{dy}{dx} = \cos x \]\[ y' = x^2 + 2x - 5 \]

calculus

more complicated examples:

\[ \frac{d^2r}{dt^2} = -\frac{GM}{r^2} \]

Newton's Law of Gravitation

diff. eqs. are often used to model dynamical situations (changing)

\( P(t) \) is population

\[ \frac{dP}{dt} = kP \]

\( k \): constant of proportionality \( (k > 0) \)

eq says: rate of change of pop. is proportional to pop. size.

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\[ \frac{dP}{dt} = k(L - P) \]

rate of change is proportional to

if \( P < L \) (still room to grow)

then \( (k > 0) \) \( \frac{dP}{dt} > 0 \) (grow)

when \( P \) is close to \( L \), \( \frac{dP}{dt} \) is small

when \( P = L \), \( \frac{dP}{dt} = 0 \) (no change)

\( P > L \rightarrow \frac{dP}{dt} < 0 \) (decline)

Solution is a function that satisfies the diff. eq.

\[ \frac{dy}{dx} = \cos x \quad \text{has solution} \quad y = \sin x + C \]

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We will learn techniques to solve more complicated ones like \[ \frac{dP}{dt} = k(L-P) \]

Let's see how we can qualitatively understand the solution w/o solving the diff. eq.

For example, y' = y. We can't solve it like \[ \frac{dy}{dx} = \cos x \]

Sometimes we can "guess" a solution.

What does y' = y say? We want a y (y(x)) such that it is its own derivative.

What is that function? y = e^x, so is y = 2e^x, 3e^x, etc.

So, y = Ce^x

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y' = y says the slope of tangent line on a solution curve is equal to the value of the function.

Notice even if we didn't know y = Ce^x, w/ enough slopes we can still "eye ball" the solution curves.

On the horizontal line: