Separation of variables
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Flip to reveal answersWhen is a first-order ODE separable?
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All 8 Flashcards — Separation of variables
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Question
When is a first-order ODE separable?
Answer
When dy/dx can be written as f(x)·g(y) — an x-part times a y-part.
Question
How do you separate the variables?
Answer
Divide by g(y), multiply by dx: collect all y's with dy on the left, all x's with dx on the right, then integrate.
Question
How many constants of integration after separating and integrating?
Answer
Just one — put a single +C on the right-hand side.
Question
What is an initial condition used for?
Answer
To find the constant C: substitute the known point, giving the one particular solution through it.
Question
Solve dy/dx = xy (general solution).
Answer
(1/y)dy = x dx ⇒ ln|y| = x²/2 + C ⇒ y = A e^(x²/2).
Question
∫(1/(y − a)) dy = ?
Answer
ln|y − a| + C.
Question
dT/dt = −k(T − r): what is the limiting temperature as t → ∞?
Answer
T → r (room temperature), since the exponential term decays to 0.
Question
Why should you substitute the initial condition before rearranging?
Answer
The algebra is usually simpler in the un-rearranged form (e.g. ln form), reducing errors.
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Topic 5.18 hub
Differential equations (HL only)
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