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Find a particular solution to the differential equation \( y'' - 4y' = 193 \).

a) \( y = 20t + 193 \)
b) \( y = -20t + 193 \)
c) \( y = 20t - 193 \)
d) \( y = -20t - 193 \)

Answer :

Final answer:

Option b) y = -20t + 193 is the closest answer for the particular solution to the differential equation y'' - 4y' = 193. We solve for m in the equation 0 - 4m = 193, finding m = -48.25, and choose b as 193.

Explanation:

To find a particular solution to the linear differential equation y'' - 4y' = 193, we look for a solution in the form of y = mt + b, where m and b are constants. This form assumes that the particular solution is a linear function, which usually suits equations where the non-homogeneous part (in this case, 193) is not dependent on t.

First, differentiate y to find y' and y'':

  • y' = m
  • y'' = 0 (since the derivative of a constant m is zero).

Substitute y' and y'' into the equation:

0 - 4m = 193

Solve for m:

-4m = 193

m = -48.25

The choice of b does not matter for this particular differential equation because y' and y'' do not depend on it, due to the fact that derivatives of constants are 0. So, we can set b to 193 for the particular solution.

Therefore, option b) y = -20t + 193 is the closest to the correct particular solution to the given differential equation.

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Rewritten by : Barada