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Solving First-Order Differential Equations by Separation of Variables

This guide covers the techniques for solving first-order differential equations (DEs) using the method of separation of variables. We explore how to rearrange DEs and integrate both sides to find solutions. Detailed examples include hyperbolic slope fields to illustrate solution curve shapes, such as circles centered at the origin, and other exponential solutions using integration by parts. The general solutions are discussed, along with visual aids in the form of slope fields. For a comprehensive understanding, watch the video linked for step-by-step demonstrations.

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Solving First-Order Differential Equations by Separation of Variables

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  1. IB HL Mathematics Topic 9 - Option: Calculus 9.5 First-order differential equations.

  2. Solving differential equations. • By separation of variables. First order DEs with separable variables. Examples:

  3. To solve these equations we need to re-arrange them as and integrate both sides of the equation.

  4. Example 1: Solve Hyperbolic slope fields show the solution curve shape.

  5. circle, centre (0,0), radius

  6. Show that the general solution to the differential equation can be written as Example 2: Slope field shows an exponential shape

  7. using integration by parts is the general implicit solution Example 3: Solve the following differential equation is the general solution

  8. Solve the following DEs: ANSWERS:

  9. Watch the video: https://www.youtube.com/watch?v=nNHlSB6b1HU

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