Mathematics · University
A constant-forced first-order equation
For y′ + ay = b with constant a > 0, b/a is the equilibrium value.
Method and conditions
For y′ + ay = b with constant a > 0, b/a is the equilibrium value.
y(t)=b/a + (y₀−b/a)e^(−at)
Worked examples
Given y′ + 6y = 10, y(0) = 16, calculate y(4). Round the final result to 4 decimal places if needed.
Use equilibrium 10/6, then add the decaying homogeneous term (16 − 10/6)e^(−6t).
1.6667
Given y′ + 4y = 28, y(0) = 20, calculate y(1). Round the final result to 4 decimal places if needed.
Use equilibrium 28/4, then add the decaying homogeneous term (20 − 28/4)e^(−4t).
7.2381
Try it
Given y′ + 4y = 19, y(0) = 19, calculate y(6). Round the final result to 4 decimal places if needed.
Given y′ + 3y = 4, y(0) = 12, calculate y(5). Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 4.75
Use equilibrium 19/4, then add the decaying homogeneous term (19 − 19/4)e^(−4t).
- 1.3333
Use equilibrium 4/3, then add the decaying homogeneous term (12 − 4/3)e^(−3t).
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