How Can We Predict Particular Solutions for Linear Differential Equations?

[Voq]

a) y'' + 3y' - y = 3sin3x This is not homogeneous.
b) y'' + 3y' - y = 0 This is homogeneous.
I see b) is homogeneous because it equals to 0. What are further conclusions for that.

How we can predict particular solution in a) to be: y = Asin3x + Bcos3x? And how to predict solutions for other forms like sin3x + e^x or 3sin²x...
Please if you have some examples post link.

 

[Mark44]

a) y'' + 3y' - y = 3sin3x This is not homogeneous.
b) y'' + 3y' - y = 0 This is homogeneous.
I see b) is homogeneous because it equals to 0. What are further conclusions for that.

How we can predict particular solution in a) to be: y = Asin3x + Bcos3x? And how to predict solutions for other forms like sin3x + e^x or 3sin²x...

Your questions are too broad to be answered in an online forum. The topic of solving linear differential equations with constant coefficients is covered in all textbooks on ordinary differential equations. I would recommend getting a textbook or taking a class, and if you still have questions, ask one that is more specific.

Thread closed.