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Mathematics · University

First-order exponential initial-value problem

Use the initial condition after solving the homogeneous differential equation.

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Method and conditions

Use the initial condition after solving the homogeneous differential equation.

y(t) = y₀ exp(−kt)

Worked examples

Solve y′ = −0.8y with y(0) = 63, then calculate y(8). Round the final result to 4 decimal places if needed.

Separation gives y = Ce^(−0.8t). Since C = 63, substitute t = 8.

0.1047

Solve y′ = −0.2y with y(0) = 22, then calculate y(11). Round the final result to 4 decimal places if needed.

Separation gives y = Ce^(−0.2t). Since C = 22, substitute t = 11.

2.4377

Try it

  1. Solve y′ = −0.9y with y(0) = 41, then calculate y(11). Round the final result to 4 decimal places if needed.

  2. Solve y′ = −0.6y with y(0) = 68, then calculate y(4). Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 0.0021

    Separation gives y = Ce^(−0.9t). Since C = 41, substitute t = 11.

  2. 6.1688

    Separation gives y = Ce^(−0.6t). Since C = 68, substitute t = 4.

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