Before the final

Misapplying the initial condition

The habit: Put the initial time into the general solution and solve the equation it gives for the constant, which equals the initial value only when the solution at t = 0 reduces to the constant itself.

5 practice questions are built to catch it, in 2 topics.

Practise against it: 4 questions

What it looks like

A question from slope fields and Euler’s method, with the answer this slip gives beside the right one.

Standard2 marks · 4 min

Check that $y = \sqrt{t^{2} + C}$ solves $\dfrac{dy}{dt} = \dfrac{t}{y}$ for every positive constant $C$ . Then find the solution with $y(0) = 4$ , and give $y(3)$ .

With the slip: $\displaystyle \approx 3.606$
Worked through: $\displaystyle y(3) = 5$

y(0) = √C, not C: solve √C = 4 for C first. With C = 4 the curve starts at y = 2.

See the worked solution

Where it comes up

Exponential and trigonometric functions1 question
Slope fields and Euler’s method4 questions

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Which function satisfies both $\dfrac{dy}{dx} = y$ for every $x$ and $y(0) = 3$ ?
Exponential and trigonometric functionsIntroductory
02
Each graph below shows the slope field of $\dfrac{dy}{dx} = x + y$ , with the origin marked and one curve drawn on it. Which curve is the solution through $(0, 0)$ ?
Slope fields and Euler’s methodStandard
03
Consider the initial value problem below. $\displaystyle \frac{dy}{dx} = x - 2y, \qquad y(0) = 1$
Slope fields and Euler’s methodChallenging
04
A ball is dropped from rest and falls with air resistance proportional to its speed. Its downward velocity $v(t)$ , in m/s after $t$ seconds, satisfies $\displaystyle \frac{dv}{dt} = 10 - 0.5v, \qquad v(0) = 0.$
Slope fields and Euler’s methodCapstone

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