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Quiz 9: Separable differential equations

Question

Question 1

Find the solution to the differential equation

\frac{d y}{d x} = 2 y .

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

\begin{array}{l} \frac{d y}{d x} & = 2 y, \\ \int \frac{d y}{y} & = \int 2 d x, \\ ln | y | & = 2 x + C, \\ | y | & = e^{2 x + C} = B e^{2 x} w h e r e B = e^{C}, \\ y & = A e^{2 x} . \end{array}

Not correct. Choice (d) is false.

Question 2

Find the solution to the differential equation

\frac{d P}{d t} = 3 P w h e r e P = 4 w h e n t = 0 .

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

\begin{array}{l} \frac{d P}{d t} & = 3 P, \\ \int \frac{d P}{P} & = \int 3 d t, \\ ln | P | & = 3 t + C, \\ | P | & = e^{3 t + C} = A e^{3} w h e r e A = e^{C}, \\ therefore, P & = B e^{3 t} . \\ P & = B e^{0} = B = 4 . & At t=0 \\ P & = 4 e^{3 t} . & Hence \end{array}

Question 3

Find the solution to the differential equation

\frac{d y}{d x} = x y .

Not correct. Choice (a) is false.

Your answer is correct.

\begin{array}{l} \frac{d y}{d x} & = x y, \\ \int \frac{d y}{y} & = \int x d x, \\ ln | y | & = \frac{1}{2} x^{2} + C, \\ | y | & = e^{\frac{1}{2} x + C} = A e^{\frac{1}{2} x} w h e r e A = e^{C}, \\ y & = B e^{\frac{1}{2} x} . \end{array}

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 4

Find the solution to the differential equation

\frac{d P}{d t} = 4 P + 5 .

Not correct. Choice (a) is false.

Your answer is correct.

\begin{array}{l} \frac{d P}{d t} & = 4 P + 5 = 4 (P + 1.25), \\ \int \frac{d P}{(P + 1.25)} & = \int 4 d t, \\ ln | P + 1.25 | & = 4 t + C, \\ | P + 1.25 | & = e^{4 t + C} = B e^{4 t} w h e r e B = e^{C}, \\ P + 1.25 & = A e^{4 t}, \\ P & = A e^{4 t} - 1.25 . \end{array}

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 5

Find the solution to the differential equation

\frac{d P}{d t} = 2 P - 2 P t w h e r e P = 5 w h e n t = 0 .

Not correct. Choice (a) is false.

Your answer is correct.

\begin{array}{l} \frac{d P}{d t} & = P (2 - 2 t), \\ \int \frac{d P}{P} & = \int (2 - 2 t) d t, \\ ln | P | & = 2 t - t^{2} + C, \\ | P | & = e^{2 t - t^{2} + C} = A e^{2 t - t^{2}} w h e r e A = e^{C}, \\ P & = B e^{2 t - t^{2}} . \\ P & = B e^{0} = B = 5 . & At t=0 \\ P & = 5 e^{2 t - t^{2}} . & Hence \end{array}

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 6

Find the solution to the differential equation

\frac{d N}{d t} - \frac{N}{k} = 0 w h e r e k i s a c o n s t a n t .

Your answer is correct.

\begin{array}{l} \frac{d N}{d t} & = \frac{N}{k}, \\ \int \frac{d N}{N} & = \int \frac{1}{k}, \\ ln | N | & = \frac{t}{k} + C, \\ | N | & = A e^{\frac{t}{k}} w h e r e A = e^{C}, \\ N & = B e^{\frac{t}{k}} w h e r e B i s a c o n s t a n t . \end{array}

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 7

Find the family of curves which satisfy the differential equation

\frac{d y}{d x} = - \frac{x}{y} .

Your answer is correct.

\begin{array}{l} \frac{d y}{d x} & = - \frac{x}{y}, \\ \int y d y & = - \int x d x, \\ \frac{1}{2} y^{2} & = - \frac{1}{2} x^{2} + C, \\ \frac{1}{2} x^{2} & + \frac{1}{2} y^{2} = C n o t i n g t h a t C > 0 . \\ therefore, x^{2} + y^{2} & = 2 C = r^{2} \end{array}

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 8

Find the family of curves which satisfy

\frac{d y}{d x} = 3 x^{2} (y - 1) .

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

\begin{array}{l} \frac{d y}{d x} & = 3 x^{2} (y - 1), \\ \int \frac{d y}{y - 1} & = \int 3 x^{2} d x, \\ ln | y - 1 | & = x^{3} + C, \\ | y - 1 | & = A e^{x^{3}} w h e r e A = e^{C}, \\ y & = B e^{x^{3}} + 1 w h e r e B i s a c o n s t a n t . \end{array}

Question 9

Using Newton’s law of cooling, it is found that the differential equation describing the heating of a potato in an oven is

\frac{d T}{d t} = - k (T - 200) .

What is the temperature inside the oven ?

Not correct. Choice (a) is false.

Your answer is correct.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Question 10

Use Newton’s law of cooling to determine the temperature with respect to time of a potato cooked in an oven at 200

^{o}

C, where the temperature of the potato is 20

^{o}

C shortly before being placed in the oven.

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

\begin{array}{l} \frac{d T}{d t} & = - k (T - 200), \\ \int \frac{d T}{T - 200} & = \int - k d t, \\ ln | T - 200 | & = - k t + C, \\ | T - 200 | & = A e^{- k t} w h e r e A = e^{C}, \\ T & = B e^{- k t} + 200 w h e r e B i s a c o n s t a n t . \end{array}

t = 0

T = 20,

therefore

20 = B + 200

and

B = - 180,

hence

T = 200 - 180 e^{- k t} .

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Authorised by: Head, School of Mathematics and Statistics.

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Questions:

right first attempt
right
wrong

a)	$y = x^{2} + C$	b)	$y = e^{\frac{1}{2} x}$
c)	$y = A e^{2 x}$	d)	$y = \frac{1}{2} A e^{x}$

a)	$P = A e^{3 t}$	b)	$P = 3 A e^{t}$
c)	$P = 12 A e^{t}$	d)	$P = 4 e^{3 t}$

a)	$y = \frac{1}{2} x^{2} + C$	b)	$y = B e^{\frac{1}{2} x^{2}}$
c)	$y = e^{\frac{1}{2} x^{2}}$	d)	$y = \frac{1}{2} B e^{x^{2}}$

a)	$P = A e^{4 t} + 5 t$	b)	$P = A e^{4 t} - 1.25$
c)	$P = 5 e^{4 t} + 5 t$	d)	$P = A e^{4 t} + 1.25 t$

a)	$P = 5 e^{2 - 2 t}$	b)	$P = 5 e^{2 t - t^{2}}$
c)	$P = 5 e^{2 - 2 t^{2}}$	d)	$P = P^{2} - P^{2} t$

a)	$N = B e^{\frac{t}{k}}$ where $B$ is a constant.	b)	$N = B e^{- k t}$ where $B$ is a constant.
c)	$N = B e^{- \frac{t}{k}}$ where $B$ is a constant.	d)	$N = B e^{k t}$ where $B$ is a constant.

a)	$x^{2} + y^{2} = r^{2}$	b)	$y = \sqrt{x^{2} + C}$
c)	$y = - \frac{1}{2} x^{2} + C$	d)	This differential equation cannot be solved using the techniques learned in this course.

a)	$y = B e^{x^{3}} - 1$ where $B$ is a constant.	b)	$y^{2} - y - x^{3} - C = 0$ where $C$ is a constant.
c)	$y^{2} - y + x^{3} - C = 0$ where $C$ is a constant.	d)	$y = B e^{x^{3}} + 1$ where $B$ is a constant.

a)	200k $^{o}$ C	b)	200 $^{o}$ C
c)	21.1 $^{o}$ C	d)	The temperature cannot be determined from this equation.

a)	$T = 20 e^{- k t} - 200$	b)	$T = 180 e^{- k t} - 200$
c)	$T = 21 - e^{- k t}$	d)	$200 - 180 e^{- k t}$

The University of Sydney - School of Mathematics and Statistics

Quiz 9: Separable differential equations

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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