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garri49 [273]
3 years ago
12

What do I do with e^y in the differential equation dy/dt =(2t)/(e^y)?

Mathematics
1 answer:
Lelechka [254]3 years ago
7 0

As you said, the equation is separable:

d<em>y</em>/d<em>t</em> = 2<em>t</em> / exp(<em>y</em>)

exp(<em>y</em>) d<em>y</em> = 2<em>t</em> d<em>t</em>

Integrate both sides:

∫ exp(<em>y</em>) d<em>y</em> = ∫ 2<em>t</em> d<em>t</em>

exp(<em>y</em>) + <em>C</em>₁ = <em>t</em> ² + <em>C</em>₂

Move the constant terms to one side. When you add them together, you get another constant, so you can ignore the subscript altogether:

exp(<em>y</em>) = <em>t</em> ² + <em>C</em>

Solve for <em>y</em> explicitly by taking the logarithm of both sides:

ln(exp(<em>y</em>)) = ln(<em>t</em> ² + <em>C </em>)

<em>y</em> = ln(<em>t</em> ² + <em>C </em>)

<em>C</em> can be any number; if it happens to be 0, then you have

<em>y</em> = ln(<em>t</em> ²) = 2 ln(<em>t</em> )

so B is the correct choice.

You can also approach this from the opposite angle: Assume <em>y</em> is one of the given solutions, then substitute it into the ODE. (Bit more trial-and-error involved, so not a good idea if you're short on time.)

For example, if <em>y</em> = 2 exp(<em>t</em> ) as in choice A, you have d<em>y</em>/d<em>t</em> = 2 exp(<em>t</em> ), so the ODE would become

2 exp(<em>t</em> ) = 2<em>t</em> / exp(2 exp(<em>t</em> ))

which is clearly (I hope) not true.

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If 12,025 was invested at 3.25% compounded monthly 3 years, how much interest will be earned
wariber [46]

Answer:

$1,229.75

Step-by-step explanation:

Lets use the compound interest formula provided to solve this:

A=P(1+\frac{r}{n} )^{nt}

<em>P = initial balance</em>

<em>r = interest rate (decimal)</em>

<em>n = number of times compounded annually</em>

<em>t = time</em>

<em />

First, change 3.25% into a decimal:

3.25% -> \frac{3.25}{100} -> 0.0325

Since the interest is compounded monthly, we will use 12 for n. Lets plug in the values now:

A=12,025(1+\frac{0.0325}{12})^{12(3)}

A=13,254.75

Lastly, subtract A from P to get the interest earned:

13,254.75 - 12,025 = 1,229.75

6 0
3 years ago
I need help with 5 and 6
8_murik_8 [283]

Answer:

w = 6

RS = 13

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3w = 18

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AC = AB + BC

56 = (x + 16) + (5x + 10)

56 = 6x + 26....then rearrange the equation to form...;

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7 0
3 years ago
In Geometry class, Taryn presented a counterexample to disprove a conjecture
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Answer:If a polygon is not regular will it have any congruent angles

Step-by-step explanation:

3 0
3 years ago
A building has a concrete foundation that’s 24” wide and 36” deep at all points. How many cubic yards of concrete are necessary
Iteru [2.4K]

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Step-by-step explanation:

Just so L*W*H=24(36)(30)= 25920 cubic inches=25920 (1 in*1 in* 1 in)

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do divide 25920 by (36*36*36)

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4 0
3 years ago
a baseball and a bat cost $1.10 and if this bat costs .10 cents more than the ball. how much did the ball cost?
kari74 [83]
<u>Define x:</u>
Let the cost of the ball be x.
Ball = x
Bat = x + 0.1

<u>The total cost is $1.10:</u>
x + x + 0.1 = 1.1
2x + 0.1 = 1.1
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<u>Find the cost of the bat and the ball:</u>
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Answer: The ball cost $0.50


8 0
3 years ago
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