Find the value of "x".

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find

9966 [12]

Answer:

The particular integral of given differential equation

$y_{p} = \frac{1}{4} ( x - (\frac{-5}{4} ) (1))$

General solution of given differential equation

$y = y_{c} + y_{p}$

$Y (x) = C_{1} e^{x} + C_{2} e^{4x} + \frac{1}{4} ( x + (\frac{5}{4} ))$

Step-by-step explanation:

Step(i):-

Given Differential equation y'' − 5 y' + 4 y = x

Given equation in operator form

D²y - 5 Dy + 4 y = x

⇒ ( D² - 5 D + 4 ) y =x

⇒ f(D) y = Q

where f(D) = D² - 5 D + 4 and Q(x) = x

The auxiliary equation f(m) =0

m²-5 m + 4 =0

m² - 4 m - m + 4 =0

m ( m -4 ) -1 ( m-4) =0

(m - 1) =0 and ( m-4) =0

m = 1 and m =4

The complementary function

$Y_{c} = C_{1} e^{x} + C_{2} e^{4x}$

Step(ii):-

particular integral

Particular integral

$y_{p} = \frac{1}{f(D)} Q(x) = \frac{1}{D^{2} - 5 D + 4} X$

taking common '4'

$= \frac{1}{4(1 + (\frac{D^{2} - 5 D}{4} ))} X$

$=\frac{1}{4} (1 + (\frac{D^{2} -5D}{4})^{-1} )} X$

applying binomial expression

( 1 + x )⁻¹ = 1 - x + x² - x³ +.....

$=\frac{1}{4} (1 - (\frac{D^{2} -5D}{4}) +((\frac{D^{2} -5D}{4})^{2} -...} )X$

Now simplifying and we will use notation D = $\frac{dy}{dx}$

$=\frac{1}{4} (x - (\frac{D^{2} -5D}{4})x +((\frac{D^{2} -5D}{4})^{2}(x) -...}$

Higher degree terms are neglected

$=\frac{1}{4} (x - (\frac{ -5 D}{4}) x)$

The particular integral of given differential equation

$y_{p} = \frac{1}{4} ( x - (\frac{-5}{4} ) (1))$

Final answer:-

General solution of given differential equation

$y = y_{c} + y_{p}$

$Y (x) = C_{1} e^{x} + C_{2} e^{4x} + \frac{1}{4} ( x + (\frac{5}{4} ))$

4 0

3 years ago

A garrison of 400men had food for 40days.After 10 days,200 more men joined them .How long will the food last now?(Assume that th

mestny [16]

Let $x$ be the amount of food needed for one man for one day. At the beginning, we have enough food for 400 men for 40 days, i.e.

$400\cdot 40\cdot x = 16000x$

After 10 days, the 400 men will have consumed

$400\cdot 10\cdot x = 4000x$

food, impliying that the food remaining is

$16000x-4000x=12000x$

If 200 more men join the garrison, there are now 600 men. Each of them requires $x$ food each day, and there are $12000x$ units of food remaining. They will last

$\dfrac{12000x}{600}=20$ days.

6 0

4 years ago

Se deseneaza o dereapta si pe ea trei puncte A,B si C astfel incat C se afla intre A si B. Atunci numarul unghiurilor improprii

jeyben [28]

No Spanish but can I have Brainly

5 0

3 years ago

Miami Senior high's Beta club charges $4 per person for admission to a play. The club borrowed $400 to pay for

NeTakaya

Answer:

y= 4x-400

Step-by-step explanation:

The club charges $4 per person, which would be 4x, x being the number of people. Then, there would be a subtraction from the value of 4x because the club borrowed $400, they will have to deduct that from the amount the earn.

7 0

3 years ago

Un cometa es visible desde la tierra cada 24 años y otro, cada 36 años. El último año que fueron visibles conjuntamente fue en 1

mr_godi [17]

Answer:

Ambos cometas coincidirán en 2016.

Step-by-step explanation:

(The following exercise is presented in Spanish and for such reason explanation will be held in that language).

Sean $t_{1}$ y $t_{2}$ los años en los que se hacen visibles los cometas, respectivamente. Se sabe que los cometas coinciden cuando se pueden ver el mismo año, ese año estaría asociado al mínimo común múltiplo de ambos intervalos de tiempo, el cual sería igual a 72 años. Entonces, ambos cometas volverán a coincidir en 2016.

4 0

4 years ago