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When is a first-order differential equation 'separable'?
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All Flashcards in Topic 5.14
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5.14.18 cards
When is a first-order differential equation 'separable'?
When dy/dx can be written as f(x)·g(y) — an x-part times a y-part — so the variables can be split onto opposite sides.
What are the steps to solve a separable DE?
Separate (∫1/g(y) dy = ∫f(x) dx), integrate both sides with ONE + C, use the initial condition to find C, then make y the subject and interpret.
How many constants of integration appear when you solve a separable DE?
Just one + C — it absorbs the constant from each side; write it once on the side you integrate last.
What is the solution of dy/dx = ky?
y = A e^(kx), where A is the value at x = 0; k > 0 is growth, k < 0 is decay.
In y = A e^(kx), what does the constant A represent?
The starting value — the value of y when x = 0 (found from the initial condition).
Solve dV/dt = −2√V with V(0) = 400.
∫V^(−1/2) dV = ∫−2 dt ⇒ 2√V = −2t + C; V(0)=400 ⇒ C = 40; so √V = 20 − t and V = (20 − t)².
Newton's law of cooling dθ/dt = −k(θ − r): what is the long-term temperature?
θ → r, the room temperature, as t → ∞ (the exponential term decays to 0). The solution is θ = r + A e^(−kt).
Why is a pure exponential growth model dN/dt = kN unrealistic for large N?
It grows without any limit; real populations are capped by resources, so a logistic model (rate ∝ N(M − N)) is needed once N is large.
Topic 5.14 study notes
Full notes & explanations for Differential equations (HL only)
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