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Marrrta [24]
3 years ago
8

Find a particular solution to the nonhomogeneous differential equation y'' + 4 y = cos(2x) + sin(2x).

Mathematics
1 answer:
alukav5142 [94]3 years ago
6 0
The characteristic solution follows from solving the characteristic equation,

r^2+4=0\implies r=\pm2i

so that

y_c=C_1\cos2x+C_2\sin2x

A guess for the particular solution may be a\cos2x+b\sin2x, but this is already contained within the characteristic solution. We require a set of linearly independent solutions, so we can look to

y_p=ax\cos2x+bx\sin2x

which has second derivative

{y_p}''=(-4ax+4b)\cos2x+(-4bx-4a)\sin2x

Substituting into the ODE, you have

y''+4y=\cos2x+\sin2x
\implies4b\cos2x-4a\sin2x=\cos2x+\sin2x
\implies\begin{cases}4b=1\\-4a=1\end{cases}\implies a=-\dfrac14,b=\dfrac14

Therefore the particular solution is

y_p=-\dfrac14x\cos2x+\dfrac14x\sin2x

Note that you could have made a more precise guess of

y_p=(a_1x+a_0)\cos2x+(b_1x+b_0)\sin2x

but, of course, any solution of the form a_0\cos2x+b_0\sin2x is already accounted for within y_c.
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If a system of equations has infinite solutions, what do you know about the graph?
Anton [14]

Answer:

C

Step-by-step explanation:

A: Perpendicular lines have 1 solution. A is not the answer.

B: If the lines intersect, there is one solution, not many.

C: The lines are the same or can be reduced to being the same. C is the answer.

D: The lines don't intersect means that the lines are parallel. There isn't a solution.

3 0
3 years ago
In the grade 9, the ratio of girls to boys is 17:8. What percent of the grade 9 are boys?
Maurinko [17]

Answer:

32%

Step-by-step explanation:

Let's say there are 17 girls and 8 boys. So there is 25 people in total.

8/25 = 32/100=32%

3 0
3 years ago
Use the given equation to determine if the points in the brackets are on the graph
dmitriy555 [2]

The points in the brackets are on the graph are (0, 2) and (6, 10)

<h3>How to determine if the points in the brackets are on the graph?</h3>

The equation of the line is given as

4x - 3y = -6

The points are given as:

{(0,2),(1,3),(4,7),(6,10)}

Rewrite as

(x, y) = {(0,2),(1,3),(4,7),(6,10)}

Next, we substitute the x and y values in the equation 4x - 3y = -6

So, we have

(0, 2):

4(0) - 3(2) = -6

-6 = -6 ---- true

(1, 3):

4(1) - 3(3) = -6

-5 = -6 ---- false

(4, 7):

4(4) - 3(7) = -6

-5 = -6 ---- false

(6, 10):

4(6) - 3(10) = -6

-6 = -6 ---- true

Hence, the points in the brackets are on the graph are (0, 2) and (6, 10)

Read more about linear equations at:

brainly.com/question/2226590

#SPJ1

3 0
1 year ago
Mr.kelly pays $12,564 a year for rent. His rent is a constant amount each month. Which equation represents the amount he pays he
Nuetrik [128]

Answer:

1047 m = c

Step-by-step explanation:

Given : Mr.kelly pays $12,564 a year for rent. His rent is a constant amount each month.

To Find:Which equation represents the amount he pays her month if m=months and c=total rent paid for the year?

Solution :

Since he pays for a year (12 months) =  $12,564

Since  His rent is a constant amount each month.

He pays for one month = \frac{12564}{12}=1047

Thus the amount he pays per month $1047

If he pays for m months then the total amount(c) of rent for m months

⇒1047 m = c

Hence equation represents the amount he pays her month if m=months and c=total rent paid for the year: 1047 m = c

3 0
3 years ago
Which strategy is the most appropriate strategy to solve 5x^2+30x ?
guapka [62]
B. Square root property
4 0
3 years ago
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