Differential Equations 101

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Questions and Answers

What are the two ways to classify differential equations?

  • By their color and size
  • By their order and degree (correct)
  • By their flavor and texture
  • By their shape and weight

What does the general solution of a differential equation satisfy?

  • The equation and initial/boundary conditions
  • Only the equation (correct)
  • Only initial/boundary conditions
  • None of the above

What does the particular solution of a differential equation satisfy?

  • Only the equation
  • Only initial/boundary conditions
  • Both the equation and initial/boundary conditions (correct)
  • None of the above

What are autonomous differential equations?

<p>Equations that do not depend explicitly on the independent variable (D)</p> Signup and view all the answers

What are homogeneous differential equations?

<p>Equations that do not have a non-zero term (B)</p> Signup and view all the answers

What are non-homogeneous differential equations?

<p>Equations that have a non-zero term (A)</p> Signup and view all the answers

What is the complementary function of a non-homogeneous differential equation?

<p>The general solution of the corresponding homogeneous differential equation (B)</p> Signup and view all the answers

What is the particular integral of a non-homogeneous differential equation?

<p>The particular solution (A)</p> Signup and view all the answers

What is the Wronskian used for?

<p>To test the linear independence of functions (A)</p> Signup and view all the answers

What is involved in series solutions of differential equations?

<p>Expressing the solution as a power series (A)</p> Signup and view all the answers

Flashcards

Classifying Differential Equations

By their order (highest derivative) and degree (power of the highest derivative).

General Solution

The general solution satisfies only the differential equation itself, containing arbitrary constants.

Particular Solution

A particular solution satisfies both the differential equation and any initial or boundary conditions.

Autonomous Differential Equations

Differential equations where the independent variable does not explicitly appear in the equation.

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Homogeneous Differential Equations

Differential equations that do not have a non-zero term; all terms involve the dependent variable or its derivatives.

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Non-Homogeneous Differential Equations

Differential equations that have a non-zero term, meaning there's a term that doesn't involve the dependent variable or its derivatives.

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Complementary Function

The general solution to the corresponding homogeneous differential equation of a non-homogeneous equation.

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What is particular integral?

A particular solution that satisfies the non-homogeneous differential equation.

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Wronskian

A determinant used to test the linear independence of a set of functions.

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Series Solutions

A method of finding a solution to a differential equation by expressing it as an infinite power series.

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Study Notes

  1. Differential equations can be classified by their order and degree.
  2. The general solution of a differential equation satisfies the equation but may not satisfy initial/boundary conditions.
  3. The particular solution of a differential equation satisfies both the equation and initial/boundary conditions.
  4. Autonomous differential equations do not depend explicitly on the independent variable.
  5. Homogeneous differential equations do not have a non-zero term.
  6. Non-homogeneous differential equations have a non-zero term.
  7. The complementary function of a non-homogeneous differential equation is the general solution of the corresponding homogeneous differential equation.
  8. The particular integral of a non-homogeneous differential equation is the particular solution.
  9. The Wronskian is a determinant used to test the linear independence of functions.
  10. Series solutions of differential equations involve expressing the solution as a power series.

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