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**Calculus summary**

The overview of this lecture is that to form a graph in geometric from ODE in analytic. By doing this, this lecture is organized into five sections. The first section is talking about First-order ODE’s, ODE is ordinary differential equation, and at view of ODE’s in geometric. And the second section is thinking at what the graph of Direction field or Integral curve in geometric is. And the third section is how to draw the graph, taking a couple of exercise how computer or human does. And the fourth section is thinking at what the principle of drawing Integral line is by using two examples. And final section is what the example which violates before section’s principle is and what the wrong of the example is and the reason why it is wrong.

About first-order ODE’s and at view of ODE’s in geometric

This section begins with explaining examples of ODE which are y’=x/y, y’=x-y2, y’=y=x2. And this section is also explaining the geometry against the analysis, so for example, y’=f(x, y) in analysis is something called Direction field in geometry, and y1(x) in analysis is something called Integral curve in geometry.

What the graph of direction field or Integral curve in geometric

The Direction field which example is f(x, y) in analysis is a slope in each element. And the Integral curve is a curve which goes to the plain in every point, changing in elements, and everywhere has the direction of field everywhere at all points. So that, for example, “y’(x) =f(x, y(x)” is same as “f(x, y1(x1))”

How to draw the graph

It’s different from computer and human how to draw the graph. Computer takes three steps to draw it. Pick up the point (x, y) equally spaced, and then find f(x, y) at the point, and then draw the slope in each required point. But when human does that, pick slope C, and then plot equation f(x, y) (which is an ordinary curve) = C in general Isocline. And then draw the Integral curve.

What the principle of drawing Integral line by using two examples

When it’s giving you an equation “y’= -x/y”, its isocline is “y= -x/c”. So putting in an Integral curve, it comes a circle that is “x2 + y2 = ”. And it can be also replaced “y = y1(x) = + ”. And it has to add “+” because of indicating a negative solution of the Integral curve.

In the other hand, when it’s giving you an equation “y’=1+x-y”, its isocline is “y = 1+x-c”. So putting in an Integral curve, both the Isocline and the Integral line are overlapping when c=1, y=x.

Through the two examples, they show two principles which are two Integral curves can’t cross and cannot be tangent. So it says that it’s just one exist solution and only one unique solution through (x0, y0). To do that, f(x, y) should be continuous near the (x0, y0).

What the wrong example and why so

If the principle is violated, what the graph indicates. For example, the equation xy’=1-y indicates the graph like a flower. It is overlapped at the point (0, 1) which is no uniqueness and the line y=0 is no existence. It’s because its equation “dy/dx = (1-y)/x” is not continuous when x=0, so it can’t be done.