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Ber [7]
3 years ago
12

Find the general solution of x'1 = 3x1 - x2 + et, x'2 = x1.

Mathematics
1 answer:
Ann [662]3 years ago
7 0
Note that if {x_2}'=x_1, then {x_2}''={x_1}', and so we can collapse the system of ODEs into a linear ODE:

{x_2}''=3{x_2}'-x_2+e^t
{x_2}''-3{x_2}'+x_2=e^t

which is a pretty standard linear ODE with constant coefficients. We have characteristic equation

r^2-3r+1=\left(r-\dfrac{3+\sqrt5}2\right)\left(r+\dfrac{3+\sqrt5}2\right)=0

so that the characteristic solution is

{x_2}_C=C_1e^{(3+\sqrt5)/2\,t}+C_2e^{-(3+\sqrt5)/2\,t}

Now let's suppose the particular solution is {x_2}_p=ae^t. Then

{x_2}_p={{x_2}_p}'={{x_2}_p}''=ae^t

and so

ae^t-3ae^t+ae^t=-ae^t=e^t\implies a=-1

Thus the general solution for x_2 is

x_2=C_1e^{(3+\sqrt5)/2\,t}+C_2e^{-(3+\sqrt5)/2\,t}-e^t

and you can find the solution x_1 by simply differentiating x_2.
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Minchanka [31]

Answer:

30

Step-by-step explanation:

30/-5=-6

due to integer rules the number would be negative

4 0
3 years ago
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2. What is the product of 4 2/3 and 11 1/4?
Lerok [7]
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VashaNatasha [74]
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3 years ago
Gabriela has dinner at a cafe and the cost of her meal is $45.00. Because of the service,she wants to leave a 15%tip. What is he
Kitty [74]
45+(45*.15)=
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5 0
3 years ago
the length of a rectangle is 3 feet more than twice the width. if the width is increased by four feet the perimeter of the new r
densk [106]

Answer:

The area of the original rectangle  = 189 sq ft.

Step-by-step explanation:

Let, the width of the rectangle = m feet

So, the length of the rectangle =  (2 m + 3)  ft

Now, new width w'  = (m + 4)

Perimeter of new rectangle (P')   = 68 ft

Perimeter of a rectangle = 2 (LENGTH + WIDTH)

⇒68 ft   = 2( L + W')  

or, 68 ft   = 2 [( 2m + 3) + (m + 4)]

or, 2 (3m + 7)  = 68

or, 3m + 7 = 68/2  = 34

⇒ 3m = 34 - 7 = 27

⇒ m = 27/3 = 9

or, m = 9 ft

Hence, the original width  of the rectangle =  9 ft

Original Length of the rectangle = 2m  + 3  = 2(9) + 3 = 21 ft

Now, Area of the Rectangle = LENGTH X WIDTH

=  21 ft  x  9 ft  = 189 sq ft

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3 years ago
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