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Introduction to Differential Equations and Their Solutions

Differential equations are equations involving derivatives of unknown functions. They may include independent variables like x or t, dependent variables such as y, and derivatives like dy/dx or dy/dt. Understanding these equations is crucial, as every anti-derivative you've encountered is a differential equation. The course covers different types, solution methods, and examples demonstrating concepts through "guess and check." Students will gain insight into the solutions of various differential equations through systematic exploration, problem-solving, and hands-on assignments.

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Introduction to Differential Equations and Their Solutions

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  1. Differential Equations (10/14/13) • A differential equation is an equation which contains derivatives within it. • More specifically, it is an equation which may contain an independent variable x (or t) and/or a dependent variable y (or some other variable name), but definitely contains a derivativey ' = dy/dx (or dy/dt). • It may also contain second derivatives y '' , etc.

  2. Examples of DE’s • Every anti-derivative (i.e., indefinite integral) you have solved (or tried to solve) this semester is a differential equation! • What is y if y ' = x2 – 3x+ 5 ? • What is y if y ' = x / (x2 + 4) • What is y if dy/dt = e0.67t • Note that you also get a “constant of integration” in the solution.

  3. New types of examples • The following is a DE of a different type since it contains the dependent variable:y ' = .08y • Say in words what this says! Sound familiar? • Note that we don’t see the independent variable at all – let’s call it t . • What is a solution to this equation? And how can we find it?

  4. Clicker Question 1 • What is the most general solution of the differential equation 2 dy/dt = 5 / t4 ? • A. -5 / (3 t 3) + C • B. 5 / (3 t 3) + C • C. -5 / (6 t 3) + C • D. -1 / t 5 + C • E. -2 / t 5 + C

  5. The solutions to a DE • A solution of a given differential equation is a functiony which makes the equation work. • Show that y = Ae0.08t is a solution to the DE on the previous slide, where A is a constant. • Interpret this result! • Note that we are using the old tried and true method for solving equations here called “guess and check”.

  6. Examples of guess and check for DE’s • Show that y = 100 – A e –t satisfies the DE y ' = 100 - y • Show that y = sin(2t) satisfies the DE d2y / dt 2 = -4y • Show that y = x ln(x) – x satisfies the DEy ' = ln(x) • Of course one hopes for better methods to solve equations, but DE’s can be very hard.

  7. Clicker Question 2 • Which function below satisfies the DEd2y/dx2 = 9y ? • A. y = 3ex • B. y = 3sin(x) • C. y = sin(3x) • D. y = x2 + 9x • E. y = e3x

  8. Assignment for Wednesday • Read Section 9.1. • On page 584, do # 1 – 7 odd.

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