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The complementary function of $\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}+2y=6e^{x}+\sin x$ is:
A$C_1e^{-x}+C_2e^{-2x}$
B$C_1e^{x}+C_2e^{2x}$
C$C_1e^{-x}+C_2xe^{-x}$
D$C_1\cos x+C_2\sin x$
Answer & Solution
Correct answer: A. $C_1e^{-x}+C_2e^{-2x}$
1. The complementary function solves the homogeneous equation $y''+3y'+2y=0$.
2. Form the auxiliary equation $m^2+3m+2=0$.
3. Factorising: $(m+1)(m+2)=0$, so $m=-1$ or $m=-2$.
4. Two distinct real roots give $y_c=C_1e^{-x}+C_2e^{-2x}$.
5. The positive exponents come from misreading the signs of the roots; a repeated root would be needed for the $xe^{-x}$ form.
_Source: NECTA ACSEE 2023 Advanced Mathematics 142/2, Question 7: Differential Equations_
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