Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 526: 47

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We can notice that: $F(x)=\sin^2 x \cos^2 x$ Obtain: $(D^2+6)y_p(x)=\sin^2 x \cos^2 x$ Therefore, the general solution for the given differential equation is: $y(x)=c_1\cos 3x+c_2\sin 3x+A_0+A_1\cos 4x +A_2\sin 4x$ The trial solution is $y_p=A_0+A_1\cos 4x +A_2\sin 4x$.