Lec 16 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by addressing a correction made to a previous formula, emphasizing the importance of accuracy in mathematical communication. They then introduce the topic of differential equations, highlighting the relevance of this subject to various fields, including quantum mechanics. The focus is on a specific type of differential equation where the rate of change of x with respect to y is equal to a function f(x). The professor simplifies the solution approach by suggesting that antiderivatives, specifically through substitution or advanced guessing, are sufficient for the scope of the class. The session concludes with an introduction to a simple yet significant differential equation involving the annihilation operator, which is foundational in quantum mechanics.

This paragraph delves into solving a differential equation that describes the ground state of a harmonic oscillator. The equation involves both x and y, making it more complex than previous examples. The professor employs a clever technique involving differentials to separate the variables, allowing for the equation to be solved by integrating both sides. The solution process involves taking antiderivatives, leading to an expression involving the natural logarithm and a quadratic function of x. The final solution is expressed in terms of an exponential function, which is recognized as the normal distribution, a fundamental concept in probability and quantum mechanics. The professor also ensures that the solution is checked for accuracy by differentiating it back and verifying that it satisfies the original equation.

The professor outlines the general method for solving differential equations when they can be expressed in the form of f(x)g(y). This method, known as separation of variables, involves dividing by g(y) and multiplying by dx, effectively separating the differentials and allowing for integration of both sides. The process results in an implicit equation that defines y in terms of x, which can sometimes be converted into an explicit equation. The professor also discusses the handling of constants and the importance of considering both positive and negative values for y, as well as the special case when y equals zero. The commentary highlights the importance of understanding the underlying concepts and the limitations of the method, such as the exclusion of certain solutions when division by a variable is involved.

The script transitions into using differential equations to solve geometric problems. The first example involves finding a curve where the slope of the tangent line at any point is twice the slope of the line from the origin to that point on the curve. The differential equation representing this relationship is derived and solved using the separation of variables technique. The solution is an implicit equation that describes a family of curves, which are then exponentiated to find the explicit form of the solution. The professor also addresses the importance of considering the implications of the solution, such as the behavior of the curves for positive and negative values of y, and the exclusion of the a = 0 case due to the division by y in the solution process.

This paragraph further explores the properties of the solution to the geometric problem presented earlier. The professor discusses the importance of checking the solution to ensure it satisfies the original differential equation and addresses the issue of the solution being undefined at x = 0. They explain that the rate of change is undefined at this point, which means that knowing the value and the rate of change at x = 0 does not uniquely specify the function. The professor warns that the function could potentially transition from one branch to another across x = 0, suggesting that the solution should be visualized as rays emanating from the origin rather than complete parabolas.

The professor presents a fourth example involving curves perpendicular to parabolas. They derive the differential equation for the slope of these curves, which is the negative reciprocal of the slope of the tangent to the parabola. The differential equation is solved using the separation of variables technique, resulting in an implicit equation that describes a family of ellipses. The professor notes that not all constants are valid in this solution and only positive values of the constant are permissible. The explicit solutions are also provided, revealing that the ellipses are only partially valid solutions due to the vertical slope at y = 0, where the solution stops. The professor emphasizes the importance of being cautious with these solutions and understanding their geometric implications.

In the final paragraph, the professor reviews the material covered in the lesson and provides guidance for an upcoming test. They reiterate the main theme of the unit, which is that information about the derivative, and sometimes the second derivative, can provide information about the function itself. This concept is central to the study of ordinary differential equations. The professor encourages students to review the material and to look at previous tests to understand the types of problems that will be asked. They conclude by emphasizing the importance of this foundational knowledge in the study of differential equations.