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romanna [79]
3 years ago
10

Find a solution to y'(t) = te^-t satisfying the condition y(1) = 1.

Mathematics
1 answer:
Alex_Xolod [135]3 years ago
4 0

Answer:

y=-e^{-t}(t+1)+1+\frac{2}{e}

Step-by-step explanation:

The given differential equation is

y'(t)=te^{-t}

It can be written as

\frac{dy}{dt}=te^{-t}

dy=te^{-t}dt

Integrate both sides.

\int dy=\int te^{-t}dt

Apply ILATE rule on right side. Here, t is first function and e^{-t} is the second function.

y=t\int e^{-t}-\int (\frac{d}{dt}t\int e^{-t})

y=-te^{-t}-\int (1\times (-e^{-t}))         \int e^{-x}=-e^{-x}+C

y=-te^{-t}+\int e^{-t}

y=-te^{-t}-e^{-t}+C             .... (1)

Initial condition is y(1) = 1. It means at t=1 the value of y is 1.

1=-(1)e^{-t}-e^{-(1)}+C

1=-e^{-1}-e^{-1}+C

1=-2e^{-1}+C

1=-\frac{2}{e}+C

Add \frac{2}{e} on both sides.

1+\frac{2}{e}=C

Substitute the value of C in equation (1).

y=-te^{-t}-e^{-t}+1+\frac{2}{e}

y=-e^{-t}(t+1)+1+\frac{2}{e}

Therefore, the solution of given initial value problem is y=-e^{-t}(t+1)+1+\frac{2}{e}.

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2 years ago
What fraction equal to 1/2 with a denominator of 6 and a fraction equal to 1/3 with a denominator of 6. Then add the fractions.
horrorfan [7]

Answer:

5/6

Step-by-step explanation:

1/2 + 1/3 need a common denominator to add them up.

1/2 is the same as 3/6.

1/3 is the sane as 2/6.

3/6 + 2/6

Add the tops, but keep the same denominator.

3/6 + 2/6 = 5/6

5/6 cannot be simplified, so that's the answer. That's how you do it! Hope this helps!

7 0
2 years ago
2.30
Paha777 [63]
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3 years ago
At the Souvenir Shop, gemstone souvenirs cost $11.60 for 5. Which proportion can be used to find the cost of just 1 gemstone?
Natasha2012 [34]

Answer:

see below

Step-by-step explanation:

$11.60 for $5

1. Find the rate.

$11.60 divided by $5= $2.32

2. 1 gemstone= $2.32

Your anwser should be 1 over $2.32 or $2.32 over 1 depending on how you want it.

5 0
3 years ago
Which of the below descriptions shows a possible HAMILTON PATH?
ra1l [238]

Option B: FECBAD represents the Hamilton path

Explanation:

The vertices in the given graph are A,B,C,D,E and F

We need to determine the Hamilton path of the given graph.

By definition, we know that a Hamilton path touches each and every vertex in a graph exactly once.

Hence, we need to connect the vertices in such a way that the graph touches each and every vertex exactly once.

Option A: EFADECBA

From this description, we can see that the path starts from the vertex E and connects all the vertices but some of the vertices are repeated twice.

Hence, the path EFADECBA is not a Hamilton path.

Therefore, Option A is not the correct answer.

Option B: FECBAD

From this description, we can see that the path starts from the vertex F and connects all the vertices exactly once.

Hence, the path FECBAD is the Hamilton path.

Therefore, Option B is the correct answer.

Option C: ADEFBC

From this description, we can see that the path starts from the vertex A and connects all the vertices but the path from F to B has to touch the vertex A. Thus, the vertices are repeated twice.

Hence, the path ADEFBC is not a Hamilton path.

Therefore, Option C is not the correct answer.

Option D: ADECBAFE

From this description, we can see that the path starts from the vertex A and connects all the vertices but some of the vertices are repeated twice.

Hence, the path ADECBAFE is not a Hamilton path.

Therefore, Option D is not the correct answer.

6 0
3 years ago
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