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3 years ago

Find a general solution of y" + 8y' + 16y=0.

Mathematics

1 answer:

Misha Larkins [42]3 years ago

6 0

Answer:

The general solution: $C_{1}e^{-4x} + xC_{2}e^{-4x}$

Step-by-step explanation:

Differential equation: y'' + 8y' + 16y = 0

We have to find the general solution of the above differential equation.

The auxiliary equation for the above equation can be writtwn as:

m² + 8m +16 = 0

We solve the above equation for m.

(m+4)² = 0

$m_{1}$ = -4, $m_{2}$ = -4

Thus we have repeated roots for the auxiliary equation.

Thus, the general solution will be given by:

y = $C_{1}e^{m_{1}x} + xC_{2}e^{m_{2}x}$

y = $C_{1}e^{-4x} + xC_{2}e^{-4x}$

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Fundamental theorem of calculus<br> <img src="https://tex.z-dn.net/?f=g%28s%29%3D%5Cint%5Climits%5Es_6%20%7B%28t-t%5E4%29%5E6%7D

mr_godi [17]

Answer:

$\displaystyle g'(s) = (s-s^4)^6$

Step-by-step explanation:

The Fundamental Theorem of Calculus states that:
$\displaystyle \frac{d}{dx}\left[ \int_a^x f(t)\, dt \right] = f(x)$

Where <em>a</em> is some constant.

We can let:
$\displaystyle g(t) = (t-t^4)^6$

By substitution:

$\displaystyle g(s) = \int_6^s g(t)\, dt$

Taking the derivative of both sides results in:
$\displaystyle g'(s) = \frac{d}{ds}\left[ \int_6^s g(t)\, dt\right]$

Hence, by the Fundamental Theorem:

$\displaystyle \begin{aligned} g'(s) & = g(s) \\ \\ & = (s-s^4)^6\end{aligned}$

3 0

2 years ago

The freezing point of water is 32 f. Tim added potassium and found out that the freezing point went down by 8 f what was the fre

sattari [20]

24 degrees Fahrenheit

8 0

3 years ago

The lines y= 2x+4 and y=x-3 intersect. find the coordinates of the point of intersection

Yuri [45]

Answer:

(-7, -10)

Step-by-step explanation:

Solve for x and y in double variable equation

y = 2x + 4

y = x - 3

y = x - 3, plug in to first equation

x - 3 = 2x + 4

x = -7

y = -7 - 3

y = -10

(-7, -10)

5 0

2 years ago

The average of A and B is 6. The average of X, Y, and Z is 16. What is the average of A, B, X, Y, and Z?

ser-zykov [4K]

Answer:

Step-by-step explanation:

Average of A and B is 6

(A+B) /2 = 6

Multiply each side by 2

A+B = 12

average of X, Y, and Z is 16

(X+Y+Z)/3 = 16

Multiply each side by 3

(X+Y+Z) = 48

To find the average of all of them

(A+B+X+Y+Z)/5

replace A+B with 12 and X+Y+Z with 48

(12+48)/5

60/5 = 12

4 0

3 years ago

Express as a difference.

ElenaW [278]

10

b: 5+5
c: 5+5

thats thw andeer

3 0

2 years ago