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diamong [38]
2 years ago
9

Problem 1: Use the Laplace Transforms to solve:

Mathematics
1 answer:
Leokris [45]2 years ago
3 0

Transforming the ODE yields

L\left\{y'' - y'\right\} = L\left\{e^{-3t}\right\}

(s^2 Y(s) - sy(0) - y'(0)) - (s Y(s) - y(0)) = \dfrac1{s+3}

(s^2 - s) Y(s) = \dfrac1{s+3}

Y(s) = \dfrac1{(s^2 - s)(s+3)} = \dfrac1{s(s-1)(s+3)}

Partial fractions:

Y(s) = \dfrac as + \dfrac b{s-1} + \dfrac c{s+3}

\implies 1 = a(s-1)(s+3) + b s(s+3) + c s(s - 1)

\implies 1 = -3 a + (2 a + 3 b - c) s + (a + b + c) s^2

\implies \begin{cases}-3a=1 \\ 2a+3b-c=0 \\ a+b+c=0\end{cases} \implies a=-\dfrac13, b=\dfrac14, c=\dfrac1{12}

\implies Y(s) = -\dfrac13 \times \dfrac1s + \dfrac14 \times \dfrac1{s-1} + \dfrac1{12} \times \dfrac1{s+3}

Take the inverse transform:

\boxed{y(t) = -\dfrac13 + \dfrac{e^t}4 + \dfrac{e^{-3t}}{12}}

You might be interested in
Anybody know the answer?
marysya [2.9K]
I'd start by writing an equation for each of the right triangles. (Pythagorean theorem)

y² + 9² = z²
x² + z² = (4+9)²
4² + y² = x²

we want to find z so combine the equations by substituting the other variables x,y out.

substitute y² for (x² - 4²) in 1st equation.

(x² - 4²) + 9² = z²

now by rearranging the 2nd equation we can substitute x² for (13² - z²)

(13² - z²) - 4² + 9² = z²
169 - z² - 16 + 81 = z²
234 - z² = z²
234 = 2z²
234/2 = z²
117 = z²
√(117) = z
√(9*13) = z
3√(13) = z

13 goes in the box




































6 0
2 years ago
Round 13.406 to the nearest hundredths
hjlf
13.41 is 13.406 rounded to nearest hundreth
8 0
3 years ago
Read 2 more answers
Triangle PQR has the coordinates P(2,4), Q(-2,4), R(0,-5).
tensa zangetsu [6.8K]

Answer:

I hope I got this correct.

Step-by-step explanation:

Multiply the coordinates by 2.

P(4,8)

Q(-4,8)

R(0,-10)

For the future if you get stuck, multiply the coordinates by the dilation number.

4 0
2 years ago
Can someone help me with this. ^_^ THANKS
Fynjy0 [20]
It would be B. because it's just 7 times 1.5
4 0
3 years ago
Please help me asap.​
KonstantinChe [14]

Answer:

<T

Step-by-step explanation:

You can tell that the angles and length of the triangles are equal from the marks. For example, angle J has 2 lines on its angle so i looked for the other angle with 2 lines on it which happens to be angle S.

Comment if u dont understand

3 0
2 years ago
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