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Understanding Differential Equations and Separation of Variables

This document explores the fundamentals of differential equations, particularly focusing on the separation of variables method. It begins with the introduction of a basic differential equation derived from motion, given by dy/dt = -9.8t. The concept of separating variables is explained thoroughly, leading to the general solution of the form y = -4.9t² + C. Moreover, we provide a step-by-step guide to solving a specific differential equation with an initial condition, demonstrating the process of variable separation, finding the anti-derivative, and isolating the solution in terms of the constant C.

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Understanding Differential Equations and Separation of Variables

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  1. Differential Equations and Seperation of Variables

  2. dy = - 9.8t dt What is a differential equation? An equation that resulted from differentiating another equation. ex. v(t) = - 9.8t differential equation

  3. What is a differential equation? An equation that resulted from differentiating another equation.

  4. dt * * dt dy dy = - 9.8t = - 9.8t dy = - 9.8t dt dt dt separation of variables comes from getting similar variables on one side of equation

  5. dy = - 9.8t dt anti-derivative: y = - 4.9 t2 + C This is the solution to the differential equation!!!

  6. dy dx 1 4 2001 FR Question #6 6.(b) Find y = f(x) by solving the differential equation = y2 ( 6 - 2x ) with the initial condition f(3) = . 1) separate variables 2) take anti-derivative 3) Isolate the solution in terms of C 3) find C… if dropping absolute value from ln  note j = ± C to alleviate the issue of ln y = - value. 4) rewrite original equation.

  7. 1) separate variables 2) take anti-derivative 3) Isolate the solution in terms of C 3) find C… if dropping absolute value from ln  note j = ± C to alleviate the issue of ln y = - value. 4) rewrite original equation.

  8. Solve the differential equation using two different family points, (there will be two different solutions) End Day 1

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