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What is the solution to the equation dy/dx = -5y?

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Question: What is the solution to the equation dy/dx = -5y?

Options:

  1. y = Ce^(-5x)
  2. y = -5Ce^x
  3. y = Ce^(5x)
  4. y = 5Ce^(-x)

Correct Answer: y = Ce^(-5x)

Solution: 

This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.

What is the solution to the equation dy/dx = -5y?

Practice Questions

Q1
What is the solution to the equation dy/dx = -5y?
  1. y = Ce^(-5x)
  2. y = -5Ce^x
  3. y = Ce^(5x)
  4. y = 5Ce^(-x)

Questions & Step-by-Step Solutions

What is the solution to the equation dy/dx = -5y?
  • Step 1: Recognize that the equation dy/dx = -5y is a separable differential equation.
  • Step 2: Rewrite the equation to separate the variables: dy/y = -5 dx.
  • Step 3: Integrate both sides: ∫(1/y) dy = ∫-5 dx.
  • Step 4: The left side integrates to ln|y| and the right side integrates to -5x + C, where C is a constant.
  • Step 5: Write the equation from the integration: ln|y| = -5x + C.
  • Step 6: Exponentiate both sides to solve for y: |y| = e^(-5x + C).
  • Step 7: Rewrite e^C as a new constant, which we can call C: |y| = Ce^(-5x).
  • Step 8: Since y can be positive or negative, we can drop the absolute value: y = Ce^(-5x).
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