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Viefleur [7K]
3 years ago
9

If

Mathematics
1 answer:
Leno4ka [110]3 years ago
3 0

Answer:

\frac{s^2-25}{(s^2+25)^2}

Step-by-step explanation:

Let's use the definition of the Laplace transform and the identity given:\mathcal{L}[t \cos 5t]=(-1)F'(s) with F(s)=\mathcal{L}[\cos 5t].

Now, F(s)=\int_0 ^{+ \infty}e^{-st}\cos(5t) dt. Using integration by parts with u=e^(-st) and dv=cos(5t), we obtain that F(s)=\frac{1}{5}\sin(5t)e^{-st} |_{0}^{+\infty}+\frac{s}{5}\int_0 ^{+ \infty}e^{-st}\sin(5t) dt=\int_0 ^{+ \infty}e^{-st}\sin(5t) dt.

Using integration by parts again with u=e^(-st) and dv=sin(5t), we obtain that

F(s)=\frac{s}{5}(\frac{-1}{5}\cos(5t)e^{-st} |_{0}^{+\infty}-\frac{s}{5}\int_0 ^{+ \infty}e^{-st}\sin(5t) dt)=\frac{s}{5}(\frac{1}{5}-\frac{s}{5}\int_0^{+ \infty}e^{-st}\sin(5t) dt)=\frac{s}{5}-\frac{s^2}{25}F(s).

Solving for F(s) on the last equation, F(s)=\frac{s}{s^2+25}, then the Laplace transform we were searching is -F'(s)=\frac{s^2-25}{(s^2+25)^2}

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Answer:

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Step-by-step explanation:

Multiplying 5 by itself is the same thing as raising it to the second power:

5 \times 5 = {5}^{2}  = 25

We multiply 2 by itself two seperate times. That's the same thing as raising 2 to the second power to the second power or in simpler terms raising two to the fourth power:

2 \times 2 \times 2 \times 2 = {({2}^{2})}^{2}  =  {2}^{4}  = 16

Since only multiplication is present, we can combine the two terms to get:

5^{2}  \times  {2}^{4}

4 0
2 years ago
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fomenos

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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The problem would basically go like this.

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Answer:

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Step-by-step explanation:

4 0
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