Lec 1 | MIT 18.03 Differential Equations, Spring 2006

The speaker begins by assuming the audience has a basic understanding of differential equations and modeling from previous education. They will not be explaining what differential equations or modeling are, but will focus on first-order ordinary differential equations (ODEs). The standard form of these ODEs is discussed, along with examples. The concepts of solvability and the difference between solvable and unsolvable equations in the context of elementary functions are introduced. The lecture aims to address how to approach unsolvable equations and introduces the idea of looking at differential equations geometrically and numerically.

The paragraph explains the difference between the analytic method of solving ODEs, which involves finding explicit functions that satisfy the equation, and the geometric view, which involves direction fields and integral curves. A direction field is a visual representation where each point on the plane has a line element with a slope corresponding to the differential equation. An integral curve is a curve that at every point is tangent to the corresponding line element in the direction field. The relationship between the graph of a solution and the integral curve is established, and it is shown that drawing an integral curve geometrically is equivalent to solving the ODE analytically.

The speaker discusses how computers generate direction fields by calculating the function value at each point on a grid and then drawing line elements with the corresponding slope. The spacing of points and the length of the line elements can be adjusted, and the process is considered naive due to the straightforward computation at each point. The human approach to drawing direction fields is contrasted, emphasizing the importance of understanding the concept rather than just following a mechanical process.

The human method for drawing direction fields is described as more intelligent and efficient. Instead of starting with points, one begins with a desired slope (C) and plots the corresponding isocline, which is a curve where the slope is constant. The isoclines are drawn using dashed lines to distinguish them from the integral curves (solutions), which are drawn with solid lines. The process is demonstrated with examples, showing how to fill the plane with line elements of a specific slope to form the direction field.

The paragraph delves into finding isoclines and drawing integral curves for a given differential equation (y prime equals minus x over y). The isoclines are lines through the origin, and the integral curves are shown to be circles with centers at the origin. The solutions to the ODE are expressed in terms of square roots and are limited in domain, which is a common challenge when dealing with differential equations. The importance of understanding the domain of a solution is emphasized.

The speaker outlines two key principles regarding integral curves. First, no two integral curves can cross each other at an angle because this would imply two different slopes at the same point, which is not allowed by the direction field. Second, integral curves cannot even touch, let alone cross, each other. This is explained by the Existence and Uniqueness Theorem, which states that for a given point, there is exactly one solution to the differential equation. The theorem's hypotheses are also discussed, including the need for the function and its partial derivative with respect to y to be continuous.

The final paragraph discusses the application of the Existence and Uniqueness Theorem to a specific differential equation. It is shown that the theorem does not apply along the y-axis where the function is not defined, leading to a failure of uniqueness. The example demonstrates that the theorem must be applied carefully, ensuring that the differential equation is in the proper form and that the function meets the required conditions. The speaker also mentions Euler's method but decides to forgo a detailed explanation, instead choosing to provide an example to solidify the concepts discussed.