For $\dfrac{d^2y}{dx^2}+3\dfrac{dy}{dx}+2y=6e^{x}+\sin x$, the particular integral is:

1. Handle the two forcing terms separately. 2. For $6e^{x}$, try $y_p=Ae^{x}$. Then $A+3A+2A=6A=6$, so $A=1$, giving $e^{x}$. 3. For $\sin x$, try $y_p=p\cos x+q\sin x$ and substitute. 4. Matching coefficients yields $p=-\dfrac{3}{10}$ and $q=\dfrac{1}{10}$. 5. Adding the two parts gives $e^{x}+\dfrac{1}{10}(-3\cos x+\sin x)$. Keeping the coefficient 6 skips solving for $A$. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/2, Question 7: Differential Equations_