Introduction to Differential Equations

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Questions and Answers

What is the differential equation obtained after eliminating the arbitrary constants 'A' and 'B' from the equation: y = Acos(x) + Bsin(x)?

  • dy/dx - y = 0
  • dy/dx + y = 0
  • d²y/dx² + y = 0 (correct)
  • d²y/dx² - y = 0

What is the differential equation obtained by eliminating the arbitrary constant 'A' from the equation: y = Ae^(2x) + 3?

  • dy/dx = y - 6
  • dy/dx = 2y - 6 (correct)
  • dy/dx = 2y + 6
  • dy/dx = y + 6

What is the differential equation obtained by eliminating the arbitrary constant 'C' from the equation: y = x² + Cx + 1?

  • dy/dx = x + C
  • dy/dx = 2x (correct)
  • dy/dx = 2x + C
  • dy/dx = 2x - C

Which of the following steps are involved in eliminating the arbitrary constants from an equation to form a differential equation? (Select all that apply)

<p>Solve for one or more arbitrary constants in terms of the dependent variable and its derivatives (A), Differentiate the equation repeatedly with respect to the independent variable (B), Substitute the expressions for the arbitrary constants back into the original equation (C)</p>
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What is the differential equation obtained after eliminating the arbitrary constants 'A' and 'B' from the equation: y = Ae^(3x) + Be^(2x)?

<p>d²y/dx² - 5dy/dx + 6y = 0 (B)</p>
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Eliminating arbitrary constants from an equation involving a function and its derivatives is important for:

<p>Expressing the relationship between the function and its derivatives in a concise form (A)</p>
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Which of the following equations could be used to eliminate the arbitrary constant 'A' from the equation: y = A*sin(x)?

<p>dy/dx = A*cos(x) (C)</p>
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Flashcards

Differential Equation

An equation that relates a function to its derivatives.

Arbitrary Constants

Constants in an equation that can take various values.

Implicit Differentiation

Differentiating an equation to reduce the number of constants.

Higher-Order Derivatives

Derivatives taken multiple times of a function.

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Linear Equations

Equations that graph as straight lines, often with arbitrary constants.

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Example with Two Constants

y = A * e^(mx) + B * e^(-mx) showcases two arbitrary constants.

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Solving for Constants

Expressing constants in terms of variables and substituting back.

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Simplifying After Substitution

Combining terms and simplifying the equation after substitution.

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Study Notes

Introduction to Differential Equations

  • Differential equations express the relationship between a function and its derivatives.
  • They are fundamental in fields like physics, engineering, and biology for modeling various systems.
  • Finding differential equations often involves eliminating arbitrary constants from equations involving these functions.

Methods for Eliminating Arbitrary Constants

  • Implicit Differentiation: Repeatedly differentiate the equation with respect to the independent variable (often 'x').
  • Each differentiation reduces the number of arbitrary constants by one.
  • Solving for Arbitrary Constants:
    • Express one or more arbitrary constants in terms of the dependent variable, the independent variable, and their derivatives.
    • Substitute these expressions back into the original equation to derive the desired differential equation.

Example 1: Equation with Two Arbitrary Constants

  • Consider the equation: y = A * e^(mx) + B * e^(-mx).
  • Differentiating once with respect to x: dy/dx = Ame^(mx) - Bme^(-mx)
  • Differentiating again with respect to x: d²y/dx² = Am²e^(mx) + Bm²e^(-mx)
  • Notice how the expressions for y and its derivatives relate to one another.
  • Isolate one arbitrary constant (e.g., A or B) from the first equation or the derivative, then substitute into the second equation.

Example 2: Equation with One Arbitrary Constant

  • Consider the equation: y = x² + 2Ax + B
  • Differentiate first with respect to x: dy/dx = 2x + 2A.
  • The second derivative, if calculated, would no longer contain the arbitrary constant B.
  • Solve for A: A = (dy/dx - 2x)/2
  • Substitute A back into the original equation to find the relation.
  • Substitute and Simplify:
  • The resultant equation will be the differential equation.

General Cases for Eliminating Arbitrary Constants

  • Linear equations: Equations involving functions like polynomials. These are important in engineering for modelling physical systems.
  • Polynomial equations: Techniques for handling arbitrary constants depend on the polynomial function properties.

Essential Considerations

  • Number of Constants and Derivatives: The number of arbitrary constants should equal the number of differentiations performed.
  • Higher-Order Derivatives: Situations with complex relationships might demand higher-order derivatives to form the differential equation.
  • Consistency Checks: Verify the initial equation by substituting it back into the obtained differential equation to confirm accuracy.

Summary

  • Eliminating arbitrary constants from equations involving functions and their derivatives results in differential equations.
  • Implicit differentiation is a widely used technique for this process.
  • Strategies differ based on the number of arbitrary constants.
  • Ensure the number of derivatives matches the number of unknowns for an accurate final differential equation.

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