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Ierofanga [76]
3 years ago
15

For which equations below is x = –3 a possible solution? Check all that apply.

Mathematics
1 answer:
Tju [1.3M]3 years ago
3 0
What are the options
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Find the function y1 of t which is the solution of 121y′′+110y′−24y=0 with initial conditions y1(0)=1,y′1(0)=0. y1= Note: y1 is
strojnjashka [21]

Answer:

Step-by-step explanation:

The original equation is 121y''+110y'-24y=0. We propose that the solution of this equations is of the form y = Ae^{rt}. Then, by replacing the derivatives we get the following

121r^2Ae^{rt}+110rAe^{rt}-24Ae^{rt}=0= Ae^{rt}(121r^2+110r-24)

Since we want a non trival solution, it must happen that A is different from zero. Also, the exponential function is always positive, then it must happen that

121r^2+110r-24=0

Recall that the roots of a polynomial of the form ax^2+bx+c are given by the formula

x = \frac{-b \pm \sqrt[]{b^2-4ac}}{2a}

In our case a = 121, b = 110 and c = -24. Using the formula we get the solutions

r_1 = -\frac{12}{11}

r_2 = \frac{2}{11}

So, in this case, the general solution is y = c_1 e^{\frac{-12t}{11}} + c_2 e^{\frac{2t}{11}}

a) In the first case, we are given that y(0) = 1 and y'(0) = 0. By differentiating the general solution and replacing t by 0 we get the equations

c_1 + c_2 = 1

c_1\frac{-12}{11} + c_2\frac{2}{11} = 0(or equivalently c_2 = 6c_1

By replacing the second equation in the first one, we get 7c_1 = 1 which implies that c_1 = \frac{1}{7}, c_2 = \frac{6}{7}.

So y_1 = \frac{1}{7}e^{\frac{-12t}{11}} + \frac{6}{7}e^{\frac{2t}{11}}

b) By using y(0) =0 and y'(0)=1 we get the equations

c_1+c_2 =0

c_1\frac{-12}{11} + c_2\frac{2}{11} = 1(or equivalently -12c_1+2c_2 = 11

By solving this system, the solution is c_1 = \frac{-11}{14}, c_2 = \frac{11}{14}

Then y_2 = \frac{-11}{14}e^{\frac{-12t}{11}} + \frac{11}{14} e^{\frac{2t}{11}}

c)

The Wronskian of the solutions is calculated as the determinant of the following matrix

\left| \begin{matrix}y_1 & y_2 \\ y_1' & y_2'\end{matrix}\right|= W(t) = y_1\cdot y_2'-y_1'y_2

By plugging the values of y_1 and

We can check this by using Abel's theorem. Given a second degree differential equation of the form y''+p(x)y'+q(x)y the wronskian is given by

e^{\int -p(x) dx}

In this case, by dividing the equation by 121 we get that p(x) = 10/11. So the wronskian is

e^{\int -\frac{10}{11} dx} = e^{\frac{-10x}{11}}

Note that this function is always positive, and thus, never zero. So y_1, y_2 is a fundamental set of solutions.

8 0
2 years ago
Charles and his brother are sharing the cost of a video game. The video game costs $30.70. If Charles saved $20 to buy the game
Norma-Jean [14]

hi can we be friends? if you like it do

5 0
2 years ago
At 1 p.m., there were 257 visitors in the museum.
trapecia [35]

Use function notation to represent this statement:

At 1 p.m., there were 257 visitors in the museum.

Answer:  V(4)=257

Explanation: It says that the Museum opens at 9 so it would be 4 hours. And the 257 is the number of visitors at that time.

5 0
2 years ago
Which geometric figure are always paraelograms
kolbaska11 [484]
I don’t know I hope u get it
3 0
2 years ago
In the inequality 1/ 2 x - 6 > 10, x represents Todd's age. Which phrase most accurately describes Todd's age? A) Todd is old
Tom [10]

Answer:

Option A) Todd is older than 32

Step-by-step explanation:

we have

\frac{1}{2}x-6>10

Solve for x

Adds 6 both sides

\frac{1}{2}x>10+6\\\\\frac{1}{2}x>16

Multiply by 2 both sides

x>32

so

Todd's age is greater than 32

therefore

Todd is older than 32

8 0
2 years ago
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