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ddd [48]
3 years ago
8

Consider the differential equation y'' − y' − 30y = 0. Verify that the functions e−5x and e6x form a fundamental set of solution

s of the differential equation on the interval (−[infinity], [infinity]). The functions satisfy the differential equation and are linearly independent since the Wronskian W (e−5x, e6x) = ___________?
Mathematics
1 answer:
Alex Ar [27]3 years ago
7 0

Answer:

Step-by-step explanation:

We have to take the derivatives for both functions and replace in the differential equation. Hence

for y=e^{-5x}:

y(x)=e^{-5x}\\y'(x)=-5e^{-5x}\\y''(x)=25e^{-5x}\\

for y=e^{6x}:

y(x)=e^{6x}\\y'(x)=6e^{6x}\\y''(x)=36e^{6x}\\

Now we replace in the differential equation  y'' − y' − 30y = 0

for y=e^{-5x}:

25e^{-5x}+5e^{-5x}-30e^{-5x}=0\\25+5-30=0

for y=e^{6x}:

36e^{6x}-6e^{6x}-30=0\\36-6+30=0

Now, to know if both function are linearly independent we calculate the Wronskian

W(f,g)=fg'-f'g

W(e^{-5x},e^{6x})=(e^{-5x})(6e^{6x})-(-5e^{-5x})(e^{6x})\neq 0

I hope this is useful for you

Best regard

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Taylor took 6 hours to drive home from college for Thanksgiving break, a total distance of 290 miles.
vredina [299]

Answer: at 50 mph, she drove for 4 hours.

at 45 mph, she drove for 2 hours.

Step-by-step explanation:

Let t represent the time that she spent driving at 50 miles per hour.

Taylor took 6 hours to drive home from college for Thanksgiving break. This means that the time that she spent driving at 45 miles per hour is (6 - t) hours.

Distance = speed × time

Distance covered while driving 50 miles per hour is

50t

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Since the total distance that she drove is 290 miles, it means that

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50t + 270 - 45t = 290

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At 45 miles per hour, she drove at

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B is correct, and D is correct.

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10) The figure below is a regular octagon. What is the least number of degrees that the octagon must be rotated clockwise around
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