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Mathematics

1 answer:

dimaraw [331]2 years ago

7 0

Answer:

Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate

10%. The second car depreciates at an annual rate of 15%. What is the approximate difference in the ages of the two

cars?

Step-by-step explanation:

the answer is in the question

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Set up but do not solve for the appropriate particular solution yp for the differential equation y′′+4y=5xcos(2x) using the Meth

taurus [48]

Answer:

So, solution of the differential equation is

$y(t)=-\frac{5x^2}{4}\cot 2x\cdot \cos 2x+c_1e^{-2it}+c_2e^{2it}\\$

Step-by-step explanation:

We have the given differential equation: y′′+4y=5xcos(2x)

We use the Method of Undetermined Coefficients.

We first solve the homogeneous differential equation y′′+4y=0.

$y''+4y=0\\\\r^2+4=0\\\\r=\pm2i\\\\$

It is a homogeneous solution:

$y_h(t)=c_1e^{-2i t}+c_2e^{2i t}$

Now, we finding a particular solution.

$y_p(t)=A5x\cos 2x\\\\y'_p(t)=A5\cos 2x-A10x\sin 2x\\\\y''_p(t)=-A20\sin 2x-A20x\cos 2x\\\\\\\implies y''+4y=5x\cos 2x\\\\-A20\sin 2x-A20x\cos 2x+4\cdot A5x\cos 2x=5x\cos 2x\\\\-A20\sin 2x=5x\cos 2x\\\\A=-\frac{x}{4} \cot 2x\\$

we get

$y_p(t)=A5\cos 2x\\\\y_p(t)=-\frac{5x^2}{4}\cot 2x\cdot \cos 2x\\\\\\y(t)=y_p(t)+y_h(t)\\\\y(t)=-\frac{5x^2}{4}\cot 2x\cdot \cos 2x+c_1e^{-2it}+c_2e^{2it}\\$

So, solution of the differential equation is

$y(t)=-\frac{5x^2}{4}\cot 2x\cdot \cos 2x+c_1e^{-2it}+c_2e^{2it}\\$

7 0

3 years ago

Use technology to determine an appropriate model of the data. (-1,0), (1,-4), (2,-3), (4,5), (5,12)

Vinil7 [7]

By graphing the points, The best mode of the data will be the quadratic function

There is 5 steps to find the quadratic function that models the data.

First step           ⇒ Plot the data pairs (x, y) in a coordinate plane.
                              which are the blue circles in the attached figure

Second step     ⇒ Draw the parabola you think best fits the data.
                             Which is the red function in the attached figure.

Third step         ⇒ Estimate the coordinates of three points on the
                             parabola, which are (-1, 0), (2, -3), and (5, 12).

Fourth step       ⇒ Substitute the coordinates of the points into the model
                            y = ax² + bx + c
                            to obtain a system of three linear equations.

                            a - b + c = 0           ---------(1)
                            4a + 2 b + c = -3    ---------(2)
                            25a + 5 b + c = 12 ---------(3)

Fifth step   ⇒ Solve the linear system.
                      By using the calculator
                      a = 1 , b = -2 , c = -3

So, the quadratic model for the data is ⇒ y = x² - 2x - 3