Question

Use method of undetermined coefficients to find a particular solution of the differential equation ?′′ + 9? = cos3? + 2. Check that the obtained particular solution satisfies the differential equation.

Answer 1

The particular solution to ( d^2 y(x))/( dx^2) + 9 y(x) = cos(3 x)+2 is of the form:
y_p(x) = a_1 + a_2 x cos(3 x) + a_3 x sin(3 x)
Solve for the unknown constants a_1, a_2, and a_3:
Compute ( d^2 y_p(x))/( dx^2):
( d^2 y_p(x))/( dx^2) = ( d^2 )/( dx^2)(a_1 + a_2 x cos(3 x) + a_3 x sin(3 x))
= -9 a_2 x cos(3 x) - 6 a_2 sin(3 x) + 6 a_3 cos(3 x) - 9 a_3 x sin(3 x)
Substitute the particular solution y_p(x) into the differential equation:
( d^2 y_p(x))/( dx^2) + 9 y_p(x) = cos(3 x) + 2
-9 a_2 x cos(3 x) - 6 a_2 sin(3 x) + 6 a_3 cos(3 x) - 9 a_3 x sin(3 x) + 9 (a_1 + a_2 x cos(3 x) + a_3 x sin(3 x)) = cos(3 x) + 2
Simplify:
9 a_1 + 6 a_3 cos(3 x) - 6 a_2 sin(3 x) = 2 + cos(3 x)
Equate the coefficients of 1 on both sides of the equation:
9 a_1 = 2
Equate the coefficients of cos(3 x) on both sides of the equation:
6 a_3 = 1
Equate the coefficients of sin(3 x) on both sides of the equation:
-6 a_2 = 0
Solve the system:
a_1 = 2/9
a_2 = 0
a_3 = 1/6
Substitute a_1, a_2, and a_3 into y_p(x) = a_1 + x cos(3 x) a_2 + x sin(3 x) a_3:
y_p(x) = 1/6 x sin(3 x) + 2/9