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Question
Given the differential equation f'(x) = −2f(x) with f(0) = 3, find f(x).
Understanding
Given: f′(x) = −2f(x), find f(x) if f(0) = 3.
This is a first-order linear ODE. We solve by separating variables.
Step 1
Rewrite as: df/f = −2 dx
Step 2
Integrate: ∫ df/f = −2 ∫ dx → ln|f| = −2x + C
Step 3
Solve: f(x) = Ae⁻²ˣ, where A = eᶜ
Answer
f(0) = 3 → A = 3 → f(x) = 3e⁻²ˣ
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