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Zepler [3.9K]
3 years ago
14

Solve the differential equation by variation of parameters. y''+y=secx

Mathematics
1 answer:
suter [353]3 years ago
8 0

Answer:y=

Acosx+Bsinx +cosx ln(cosx)+x sinx

Step-by-step explanation:

given  equation y''+y=secx

auxiliary  equation

p^2+1=0\\p=\pm i

so CF is y=Acosx+Bsinx

now

y_1(x)=cosx \ \ \ \ y_2(x)=sinx\\{y_1}'(x)=-sinx \ \ \ \ {y_2}'(x)=cosx

using wronskian formula

W=\begin{vmatrix}cosx &sinx \\ -sinx & cosx\end{vmatrix}

          =cos^2x+sin^2x=1

now f(x)=secx

u=-\int \frac{f(x)y_2(x)}{W(x)}dx \ and \ v=\int \frac{f(x)y_1(x)}{W(x)}dx

u=-\int \frac{secx\times sinx}{1}dx \ and \ v=\int \frac{secx\times cosx}{1}dx

u=-\int tanx dx \ and \ v=\int {1}dx

u=ln(cosx) \ \ \ and \ \ v=x

now particular integrals are

PI=cosx ln(cosx)+x sinx

total solution

y= C.F+P.I

y=Acosx+Bsinx +cosx ln(cosx)+x sinx

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Solve the following equation using the distributive property and language translation. 2(x+6)=10 ( 8th grade math )
tatuchka [14]

Answer:

X=-1

Step-by-step explanation:first you distribute 2 to x and 6 which equals 2x+12=10 then you subtract 12 from 10 which equals 2x=-2 then divide -2 by 2x to see what x equals so x=-1

6 0
2 years ago
The measures of the angles in an equilateral triangle are equal.
JulijaS [17]

Answer:

c.60°

Step-by-step explanation:

A triangle has 3 angles. Sum of angles in a triangle is 180 degrees. Three angles of an equilateral triangle are equal

So 180/3 = 60

7 0
2 years ago
At the end of 2006, the population of Riverside was 400 people. The population for this small town can be modeled by the equatio
andriy [413]

The correct format of the question is

At the end of 2006, the population of Riverside was 400 people. The population for this small town can be modeled by the equation below, where t represents the number of years since the end of 2006 and P represents the number of people. P = 400 ( 1.2 )^ t Based on this model, approximately what was the increase in the population of Riverside at the end of 2009 compared to the end of 2006?

(A) 291

(B) 691

(C) 1040

(D) 1440

Answer:

The increase in the population at the end of 2009 is 291 people

Step-by-step explanation:

We are given the equation as P = 400 ( 1.2 )^ t

where

P = No of People

t= No of Years

it is given that in the year 2006 the population is 400

this will only happen when we take t= 0

so for

Year    value of t

2006-    t = 0

2007-    t  = 1

2008-    t = 2

2009     t = 3

No of people in 2009 will be

P = 400(1.2)^3

     = 400*1.728

   P  = 691.2

Since the equation represents no of people so it can't be in decimals, Therefore the population will be 691

Increase = P(2009) - P(2006)

               = 691 - 400

                = 291

The increase in the population at the end of 2009 is 291 people.

4 0
3 years ago
Lily is using wrapping paper to cover a box with dimensions 7 centimeters by 10 centimeters by 5 centimeters. What is the least
Leto [7]
Answer: 310 centimeters square of paper

Explanation: to find the answer you need to know the surface area of the box. So you use the formula SA = 2ab+2bc+2ac
A=7
B=10
C=5
After inputting the values you get:
SA = 2(7)(10) + 2(10)(5) + 2(7)(5)
and after simplifying:
SA = 140+100+70
SA=310
4 0
3 years ago
Suppose n(x)= 22 and n(y) is 13 what is the maximum number of elements X ∪Y can have ?
Nikolay [14]
The inclusion/exclusion principle states that

|X\cup Y|=|X|+|Y|-|X\cap Y|

That is, the union has as many members as the sum of the number of members of the individual sets, minus the number of elements contained in both sets (to avoid double-counting).

Therefore, |X\cup Y| will have the most elements when the sets X and Y are disjoint, i.e. X\cap Y=\emptyset, which would mean the most we can can in this case would be

|X\cup Y|=|X|+|Y|=22+13=35

(Note that n(X)=|X| denotes the cardinality of the set X.)
3 0
3 years ago
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