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Geometrical Interpretation of Solutions

The relationship between a function and its derivatives for a first-order ODE,

$$F(\ensuremath{\frac{d{y(x)}}{d{x}}}, y(x), x) = 0$$ (19-4)

can be interpreted as a level set formulation for a two-dimensional surface embedded in a three-dimensional space with coordinates $(y', y, x)$ . The surface specifies a relationship that must be satisfied between the three coordinates.

If $y'(x)$ can be solved for exactly,

$$\ensuremath{\frac{d{y(x)}}{d{x}}} = f(x,y)$$ (19-5)

then $y'(x)$ can be thought of as a height above the $x$ -$y$ plane.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-19)

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The Geometry of First-Order ODES: Examples

Consider Newton's law of cooling that states that the rate that a body cools by radiation is proportional to the difference in temperature between the body and its surroundings:

$$\ensuremath{\frac{d{T(t)}}{d{t}}} = -k(T - T_o)$$

Make the equation simpler by converting to a non-dimensional form, let $\Theta = T/T_o$ and $\tau = t/k$ , then

$$\ensuremath{\frac{d{\Theta(\tau)}}{d{\tau}}} = (1-\Theta)$$