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Ordinary Differential Equations ( Basics )

Ordinary Differential Equations ( Basics ). A differential equation is an algebraic equation that contains some derivatives :. Recall that a derivative indicates a change in a dependent variable with respect to an independent variable .

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Ordinary Differential Equations ( Basics )

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  1. Ordinary Differential Equations (Basics) • A differential equation is an algebraic equation that contains some derivatives: • Recall that a derivative indicates a change in a dependent variable with respect to an independent variable. • In these two examples, y is the dependent variable and t and x are the independent variables, respectively.

  2. Why study differential equations? Many descriptions of natural phenomena are relationships (equations) involving the rates at which things happen (derivatives). Equations containing derivatives are called differential equations. Ergo, to investigate problems in many fields of science and technology, we need to know something about differential equations.

  3. Why study differential equations? • Some examples of fields using differentialequations in their analysis include: • Solid mechanics & motion • heat transfer & energy balances • vibrational dynamics & seismology • aerodynamics & fluid dynamics • electronics & circuit design • population dynamics & biological systems • climatology and environmental analysis • options trading & economics

  4. Examples of Fields Using Differential Equations in Their Analysis

  5. Differential Equation Basics • The order of the highest derivative ina differential equation indicates the order of the equation.

  6. Simple Differential Equations A simple differential equation has the form Its general solution is

  7. Simple Differential Equations Ex. Find the general solution to

  8. Simple Differential Equations Ex. Find the general solution to

  9. Exercise: (Waner, Problem #1, Section 7.6) Find the general solution to

  10. Example: Motion A drag racer accelerates from a stop so that its speed is 40t feet per secondt seconds after starting. How far will the car go in 8 seconds? Given: Find:

  11. Solution: Apply the initial condition: s(0) = 0 The car travels 1280 feet in 8 seconds

  12. Exercise: (Waner, Problem #11, Section 7.6) Find the particular solution to Apply the initial condition: y(0) = 1

  13. SeparableDifferential Equations A separable differential equation has the form Its general solution is Example: Separable Differential Equation Consider the differential equation a. Find the general solution. b. Find the particular solution that satisfies the initial condition y(0) = 2.

  14. Solution: a. Step 1 — Separate the variables: Step 2 — Integrate both sides: Step 3 — Solve for the dependent variable: This is the general solution

  15. Solution:(continued) b. Apply the initial (or boundary) condition, that is, substituting 0 for x and 2 for y into the general solution in this case, we get Thus, the particular solution we are looking for is

  16. Exercise: (Waner, Problem #4, Section 7.6) Find the general solution to

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