Use the method of undetermined coefficients to solve the following differential equation: y′′+4y=4x

Step-by-step explanation:

given,

y′′ + 4 y = 4 x

D² y + 4 y = 4 x

(D²+4) y = 4 x

now, writing Auxiliary  equation

m² + 4 = 0

m² = -4

m = ± 2 i

now, complimentary function

a = 0 ,  b = 2

particular integral (y_p)

y_p = a x

y'_p = a

y"_p = 0

now,

y′′+ 4 y = 4 x

0+ 4 (a x )= 4 x

4 a x = 4 x

     a = 1

now,

  y_p = x

now, general equation

answer: domain and range mean all possible values of x and y that could be substituted to the inequality. a solution means all possible values that make the inequality statement true. so while the range is still all real numbers, the solution is the shaded area, for which y will always be greater than the line of the function.