Lesson test: Separable equations

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Separable equations

1.[2p]
Why must the roots of $h (y)$ be checked separately when solving $y' = g (t) h (y)$ ?

Each root gives a constant solution that the division by $h (y)$ removes

Correct

They are the points where the initial condition fails

The answer is: Each root gives a constant solution that the division by $h (y)$ removes

They make the integral on the right side diverge

The answer is: Each root gives a constant solution that the division by $h (y)$ removes

They are the only places where the solution is defined

The answer is: Each root gives a constant solution that the division by $h (y)$ removes

2.[3p]
Which of these equations are separable?

Select all that apply

$y' = t y^{2}$

Correct

$y' = y \cos t$

Correct

$y' = t + y$

The answer is: $y' = t y^{2}$ , $y' = y \cos t$ , $y' = e^{t} / y$

$y' = e^{t} / y$

Correct

3.[2p]
A drug is cleared with a half-life of 4 hours. How many milligrams of a 300 mg dose remain after 10 hours?

CorrectNot quite: 53.0

4.[3p]
Coffee at 85 °C stands in a room at 25 °C and reads 70 °C after 6 minutes. How many minutes after pouring does it reach 50 °C?

CorrectNot quite: 18.3

5.[1p]
Separating the variables always produces an explicit formula for $y$ in terms of $t$ .

True

The answer is: False

False

Correct

6.[2p]
For the logistic equation $P' = r P (1 - P / K)$ , at which population is growth fastest?

$P = K / 2$ , where the growth rate is $r K / 4$

Correct

$P = K$ , where the population is largest

The answer is: $P = K / 2$ , where the growth rate is $r K / 4$

$P = 0$ , where there is no crowding

The answer is: $P = K / 2$ , where the growth rate is $r K / 4$

$P = K / 4$ , one quarter of the way up

The answer is: $P = K / 2$ , where the growth rate is $r K / 4$

7.[3p]
A population follows the logistic curve with $P_{0} = 100$ , $K = 1000$ and $r = 0.03$ per year. What is the population after 50 years?

CorrectNot quite: 332

8.[2p]
Put the steps of solving a separable equation in order.

Add back any constant solutions lost in the division
Divide by $h (y)$ so each side involves one variable only
Integrate both sides, adding one constant
Apply the initial condition to fix the constant

Show the answer

b, a, d, c

9.[2p]
The solution of $y' = - t / y$ with $y (0) = 3$ is $t^{2} + y^{2} = 9$ . What does that say about its interval of existence?

It exists only for $- 3 < t < 3$ , where the circle has a finite slope

Correct

It exists for all $t$ , since the equation is defined everywhere

The answer is: It exists only for $- 3 < t < 3$ , where the circle has a finite slope

It exists only for $t > 0$ , since time cannot be negative

The answer is: It exists only for $- 3 < t < 3$ , where the circle has a finite slope

It has no interval of existence, since the solution is implicit

The answer is: It exists only for $- 3 < t < 3$ , where the circle has a finite slope

1.[2p] Why must the roots of h(y) be checked separately when solving y′=g(t)h(y)?

2.[3p] Which of these equations are separable?

3.[2p] A drug is cleared with a half-life of 4 hours. How many milligrams of a 300 mg dose remain after 10 hours?

4.[3p] Coffee at 85 °C stands in a room at 25 °C and reads 70 °C after 6 minutes. How many minutes after pouring does it reach 50 °C?

5.[1p] Separating the variables always produces an explicit formula for y in terms of t.

6.[2p] For the logistic equation P′=rP(1−P/K), at which population is growth fastest?

7.[3p] A population follows the logistic curve with P0=100, K=1000 and r=0.03 per year. What is the population after 50 years?

8.[2p] Put the steps of solving a separable equation in order.

9.[2p] The solution of y′=−t/y with y(0)=3 is t2+y2=9. What does that say about its interval of existence?

1.[2p]
Why must the roots of $h (y)$ be checked separately when solving $y' = g (t) h (y)$ ?

2.[3p]
Which of these equations are separable?

3.[2p]
A drug is cleared with a half-life of 4 hours. How many milligrams of a 300 mg dose remain after 10 hours?

4.[3p]
Coffee at 85 °C stands in a room at 25 °C and reads 70 °C after 6 minutes. How many minutes after pouring does it reach 50 °C?

5.[1p]
Separating the variables always produces an explicit formula for $y$ in terms of $t$ .

6.[2p]
For the logistic equation $P' = r P (1 - P / K)$ , at which population is growth fastest?

7.[3p]
A population follows the logistic curve with $P_{0} = 100$ , $K = 1000$ and $r = 0.03$ per year. What is the population after 50 years?

8.[2p]
Put the steps of solving a separable equation in order.

9.[2p]
The solution of $y' = - t / y$ with $y (0) = 3$ is $t^{2} + y^{2} = 9$ . What does that say about its interval of existence?