Find the general solution of y”-4y’-5y=sin4t

y = -21/697 sin(4t) + 16/697 cos(4t) + C₁e^(-t) + C₂e^(5t)

Step-by-step explanation:

y" − 4y' − 5y = sin(4t)

This is a second-order, non-homogeneous equation with constant coefficients.  The general solution is the sum of the complementary and particular solutions.

First, find the complementary solution:

y" − 4y' − 5y = 0

s² − 4s − 5 = 0

(s − 5) (s + 1) = 0

s = -1 or 5

y = C₁e^(-t) + C₂e^(5t)

Next, find the particular solution.  Since g(t) = sin(4t):

y = A sin(4t) + B cos(4t)

y' = 4A cos(4t) − 4B sin(4t)

y" = -16A sin(4t) − 16B cos(4t)

Substituting:

y" − 4y' − 5y = sin(4t)

-16A sin(4t) − 16B cos(4t) − 4(4A cos(4t) − 4B sin(4t)) − 5(A sin(4t) + B cos(4t)) = sin(4t)

-16A sin(4t) − 16B cos(4t) − 16A cos(4t) + 16B sin(4t) − 5A sin(4t) − 5B cos(4t) = sin(4t)

(16B − 21A) sin(4t) − (16A + 21B) cos(4t) = sin(4t)

Match the coefficients:

16B − 21A = 1, 16A + 21B = 0

Solve the system of equations:

A = -21/16 B

16B − 21 (-21/16 B) = 1

16B + 441/16 B = 1

697B/16 = 1

B = 16/697

A = -21/697

Therefore, the general solution is:

y = -21/697 sin(4t) + 16/697 cos(4t) + C₁e^(-t) + C₂e^(5t)

50x5=$250

step-by-step explanation:

12

step-by-step explanation:

a triangle's   full angle is 180 degrees so 180 - 156 = 24. 24/2=12