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Katarina [22]
3 years ago
9

Verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a pow

er x2n + 1 let k = n + 1.] y = (-1) nx2n, (1 + x2)y' + 2xy = 0
Mathematics
1 answer:
shepuryov [24]3 years ago
6 0

Answer:

The given power series y =\sum^{\infty}_{n=0} {(-1)^n x^{2n}} is a solution of the differential equation (1+x^2)y' + 2xy = 0

Step-by-step explanation:

This is a very trivial exercise, follow the steps below for the solution:

Step 1: Since n = 0, 1, 2, 3, 4, ........, Substitute the values of n into equation (1) below.

y =\sum^{\infty}_{n=0} {(-1)^n x^{2n}}.....................(1)

y = 1 - x^2 + x^4 - x^6 + x^8.........

Step 2: Find the derivative of y, i.e. y'

y' = -2x + 4x^3 - 6x^5 + 8x^7 .............

Step 3: Substitute y and y' into equation (2) below:

(1+x^2)y' + 2xy = 0\\\\(1+x^2)(-2x + 4x^3 - 6x^5 + 8x^7......) + 2x(1 - x^2 + x^4 - x^6 + x^8.......) = 0\\\\-2x+ 4x^3 - 6x^5 + 8x^7........ - 2x^3 +4x^5 - 6x^7 + 8x^9 ......+ 2x - 2x^3 + 2x^5 - 2x^7 + 2x^9...... = 0\\\\0 = 0

(Verified)

Since the LHS = RHS = 0, the given power series y =\sum^{\infty}_{n=0} {(-1)^n x^{2n}} is a solution of the differential equation (1+x^2)y' + 2xy = 0

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A metalworker has a metal alloy that is 25% copper and another alloy that is 60% copper.How many kilograms of each alloy should
Neporo4naja [7]

Answer:

20kg of 25% copper alloy

30kg of 60% copper alloy

Step-by-step explanation:

There are 2 kinds of metal, first metal(A) have 25% copper while the second metal(B) has 60% copper. The metalworker want to create 50kg of metal, which means the total weight of both metals is 50kg (A+B = 50). The metalworker also want the metal made of a 46% which is 23kg(0.25A + 0.6B = 0.46*50). From these sentences, we can derive 2 equations. We can solve this with substitution.

A+B = 50

A= 50-B

Let's put the first equation into the second.

0.25A + 0.6B = 0.46 *50

0.25(50-B) + 0.6B =23

12.5 - 0.25B + 0.6B =23

0.35B=23 -12.5

B= 10.5/ 0.35= 30

Then we can solve A

A= 50-B

A= 50-30

A=20

8 0
3 years ago
Which statements accurately describe the function f(x) = 3(16)^3/4? Check all that apply.
uysha [10]

Answers:

These are the statements that apply:

The initial value is 3.

The range is y >0.

The simplified base is 8.

Explanation:

1) Given expression:

f(x)=3(16)^{\frac{3}{4} x

2) Check every statement:

a) The initial value is 3?

initial value ⇒ x = 0 ⇒

f(0)=3(16)^{0}=3(1)=3

∴ The statement is right.

b) The initial value is 48?

Not, as it was already proved that it is 3.

c) The domain is x > 0?

No, because the domain of the exponential functions is all the Real numbers.

d) The range is y > 0?

That is correct, the exponential function is continuous, and monotonon increasing.

The limit when x → - ∞ is zero, but y never reaches zero, and the limit when x → ∞ is + ∞, meaning that the range is y > 0.

e) The simplified base is 12?

This is how you simplify the base:

3(16)^{\frac{3}{4} x}=3{{(16}^{(3/4)})}^x=3(16^{3/4}})^{x}=3((2^4)^{3/4})^x=3(2^3)^x=3(8)^x

Which shows that the simplified base is 8 (not 12).

f) The simplified base is 8?

Yes; this was just proved.

6 0
3 years ago
Read 2 more answers
Evaluate the expression: a(b – c) if a = -8, b = 12, and c = 4
Lelu [443]
A(b-c)
First substitute values
-8(12+4)=
Next, solve using ordering of PEMDAS
-8 (12+4)=
-8 (16)=12
7 0
3 years ago
Read 2 more answers
Savannah runs a day care center. Of the 12 children at the day care center, 3 of them are
sertanlavr [38]

Answer:

\frac{1}{4}

Step-by-step explanation:

3 out of 12 children at the day care center are 5-year-olds. So, the probability of a randomly selected child being a five-year old is \frac{3}{12}, or \frac{1}{4}.

6 0
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An angle whose measure is 90° can be classified as which type of angle?
Lena [83]
Right angle is the right answer
6 0
3 years ago
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