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By using D operator or otherwise find y in term of x if: $$\frac{d^2y} {dx^2}+\frac{dy}{dx}-6y=12+50sinx$$

Using the D operator, I translated the equation into $$y(D^2+D-6)=12+50sinx$$ $$y(D+3)(D-2)=12+50sinx$$

Hence, our characteristic equation is:

$$y_c(x)=c_1e^{-3x}+c_2e^{2x}$$

Do I have to use the method of undetermined coefficients for the right-hand side $(12+50sinx)$to find our particular solution?

Is it $Acosx+Bsinx$ ?

How do I proceed from here?

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    Almost correct, choose $y_p(x) = a + b \cos x + c \sin x$. Now substitute $y_p$ into the LHS of the ODE and equate terms on each side to solve for the constants. You should find $a = -2, b = -1, c = -7$. After that $y(x) = y_h(x) + y_p(x)$. ā€“ 2017-01-13
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    Usually, operators work from the left and act on the next object to the right. That is, the application of the linear operator should look like $(D^2+Dāˆ’6)y$. ā€“ 2017-01-13
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    Nick: How is the "answer" you accepted, actually addressing the question "How do I proceed from here?" in your post? Please be specific. ā€“ 2017-01-13

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