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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Separable differential equations: separating variables, integrating both sides, and initial conditions

14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the separation method?
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For an equation dydx=f(x)g(y)\frac{dy}{dx} = f(x) g(y):
What is when separation is possible?
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A first-order ODE separates if and only if you can write dydx=f(x)g(y)\frac{dy}{dx} = f(x) g(y) for some functions ff and gg. Examples that separate:
What is the constant of integration?
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Always include a constant after integration. Usually it is cleaner to combine the constants from both sides into one CC on the right.
What is newton's law of cooling?
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A body cools toward an ambient temperature TaT_a at a rate proportional to the temperature difference:
What is implicit versus explicit solutions?
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Sometimes the integrated equation cannot be solved cleanly for yy. In that case, leave it implicit. For example,
What is with initial condition?
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Solve dydx=−2xy2\frac{dy}{dx} = -2 x y^2 with y(0)=1y(0) = 1.
What is trigonometric right side?
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Solve dydx=sin⁡xy\frac{dy}{dx} = \frac{\sin x}{y}.
What is newton's cooling?
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A cup of coffee at 90∘90^\circC is placed in a room at 20∘20^\circC. After 1010 minutes the temperature is 60∘60^\circC. Find the temperature after 3030 minutes.
What is logistic-like?
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Solve dydx=y(1−y)\frac{dy}{dx} = y(1 - y).
What is implicit form is enough?
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Solve dydx=−yx\frac{dy}{dx} = -\frac{y}{x} with y(1)=2y(1) = 2.
What is missing constant solutions?
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dydx=y2\frac{dy}{dx} = y^2 has y=0y = 0 as a constant solution that the separation step (dividing by y2y^2) would lose.
What is forgetting the constant of integration?
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Integrating both sides without a +C+ C misses an entire family of solutions.
What is algebra errors when isolating yy?
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When solving y2=f(x)+Cy^2 = f(x) + C for yy, the sign of the square root depends on continuity with the initial condition.
What is treating the constant inconsistently?
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AeCA e^C, AA, and ln⁡C\ln C all stand for arbitrary constants; pick a single name and stick with it. :::

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