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kozerog [31]
2 years ago
13

Which of the following functions are solutions of the differential equation y'' + y = sin(x)? (Select all that apply.) y = − 1

2 x cos(x)
Mathematics
1 answer:
Svetach [21]2 years ago
4 0

The function that is the solution of the differential y" + y = sin(x) is: y(x) = -1/2(x cos x)

<h3>What is a function?</h3>

A function is an expression, rule, or law in mathematics that describes a connection between one factor (the independent variable) and another variable (the dependent variable).

<h3>What is the proof of the above function?</h3>

Take a look at the the following differential equation:

y" + y = sin (x)

The auxiiliary equation is

m² + 1 = 0

m² + 1-1 = -1

m² + 0 = -1

m² = -1

m = ±√-1

m = ±i

So, the complimentary function is yₐ (x) = c₁ cos x + c₂ sin x

Let the particular integral be:

yₙ (x) = A cos x + B sin x

yₙ '(x) = - A sin x + B cos x

yₙ ''(x) = - A cos x + B cos x

yₙ ''(x) = - (A cos x + B cos x)

After we have substituted yₙ (x); and yₙ''(x) in the given differential equation

y'' + y = sin (x)

= - (A cos x + B cos x) + (A cos x + B cos x)  = sin(x)

0 = sin (x)

If we take the particular integral to be:
yₙ (x) =  x(A cos x + B cos x)

yₙ '(x) = x(-A sin x + B cos x) + A cos x + B cos x

yₙ ''(x) = x(-A cos x - B sin x) - A sin x + B cos x + -A sin x + B cos x

Substitute yₙ (x), yₙ''(x) into the stated differential equation

y'' + y = sin (x)

x (-Acosx - Bsin x) - Asinx + Bcosx + (-Asinx + Bcosx) - x (Acosx + Bsin x) = sin (x)

-Axcosx - Bxsin x - Asinx + Bcosx -Asinx + Bcosx - Axcosx + Bxsin x = sin (x)

-2Asinx + 2Bcosx = sin(x)

Compare the coefficients of like terms on both sides of the equation

-2A = 1, B = 0

A = -1/2, B = 0

Substitute A = -1/2, B =0 into the assumed solution.

yₙ(x) = x((-1/2)cosx + (0) sinx)

= -1/2xcosx +0

= -(1/2)xcosx

Now, the general solution for the given differential equation is:

y(x) = yₓ(x) +yₙ (x)

y (x) - c₁cosx + c₂sin x -1/2x cosx

Hence, the solution is:

y(x) = -1/2xcosx

Learn more about function at;
brainly.com/question/25638609
#SPJ1

Full Question:

Which of the following functions are solutions of the differential equation y'' + y = sin x? (Select all that apply.)

A) y = 1 2 x sin x

B) y = cos x

C) y = x sin x − 5x cos x

D) y = − 1 /2 x cos x

E) y = sin x

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stepan [7]

Answer:

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Step-by-step explanation:

Firstly, we write the general formula for an exponential increase as follows;

V(t) = A( 1 + r)^t

where V(t) is the value after some number of years t

A is the investment amount which is 100,000 AED

r is the rate of increase which is 16% = 16/100 = 0.16

t is the number of years which is 20

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V(20) = 100,000(1.16)^20

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