(Solved):   (3) A linear differential equation has a complementary solution \[ y_{c}=c_{1} e^{4 x}+c_{ ...

 

(3) A linear differential equation has a complementary solution \[ y_{c}=c_{1} e^{4 x}+c_{2} x e^{4 x}+c_{3}+c_{4} \sin 2 x+c_{5} \cos x+c_{6} x^{2} \] The original \( D E=f(x) \). Choose an appropriate \( y_{p} \) for each given \( f(x) \). (a) \( f(x)=6 x e^{4 x}+5 x-\sin x \) (b) \( f(x)=7 \cos 2 x-4 e^{6 x}+9 \) (c) \( f(x)=12 x^{4}+6 \sin 3 x+2 x^{2} e^{4 x} \)