Question

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

Answer 1

Answer:

$y(x) = C_1 cos 2 x + C_2 sin 2 x + x$

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

$y_c = e^{ax}(C_1 cos b x + C_2 sin b x)$

a = 0 ,  b = 2

$y_c =C_1 cos 2 x + C_2 sin 2 x$

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

$y(x) = y_c + y_p$

$y(x) = C_1 cos 2 x + C_2 sin 2 x + x$