Differential equations (separation of variables)
Practice Flashcards
Flip to reveal answersWhen is a first-order differential equation 'separable'?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 8 Flashcards — Differential equations (separation of variables)
Sign up free to track progress and get spaced-repetition review schedules.
Question
When is a first-order differential equation 'separable'?
Answer
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.
Question
What are the steps to solve a separable DE?
Answer
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.
Question
How many constants of integration appear when you solve a separable DE?
Answer
Just one + C — it absorbs the constant from each side; write it once on the side you integrate last.
Question
What is the solution of dy/dx = ky?
Answer
y = A e^(kx), where A is the value at x = 0; k > 0 is growth, k < 0 is decay.
Question
In y = A e^(kx), what does the constant A represent?
Answer
The starting value — the value of y when x = 0 (found from the initial condition).
Question
Solve dV/dt = −2√V with V(0) = 400.
Answer
∫V^(−1/2) dV = ∫−2 dt ⇒ 2√V = −2t + C; V(0)=400 ⇒ C = 40; so √V = 20 − t and V = (20 − t)².
Question
Newton's law of cooling dθ/dt = −k(θ − r): what is the long-term temperature?
Answer
θ → r, the room temperature, as t → ∞ (the exponential term decays to 0). The solution is θ = r + A e^(−kt).
Question
Why is a pure exponential growth model dN/dt = kN unrealistic for large N?
Answer
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.
Read the notes
Full study notes for Differential equations (separation of variables)
Topic 5.14 hub
Differential equations (HL only)
More from Topic 5.14
All flashcards in this topic
Math AI exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free