# Use method of undetermined coe

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

The particular solution to ( d^2 y(x))/( dx^2) + 9 y(x) = cos(3x)+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_3x 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 xsin(3 x)Substitute the particular solution y_p(x) into the differentialequation:( 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(3x) + 9 (a_1 + a_2 x cos(3 x) + a_3 x sin(3 x)) = cos(3 x) + 2Simplify: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 = 2Equate the coefficients of cos(3 x) on both sides of theequation:6 a_3 = 1Equate the coefficients of sin(3 x) on both sides of theequation:-6 a_2 = 0Solve the system:a_1 = 2/9a_2 = 0a_3 = 1/6Substitute a_1, a_2, and a_3 into y_p(x) = a_1 + x cos(3 x) a_2 + xsin(3 x) a_3:y_p(x) = 1/6 x sin(3 x) + 2/9

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