Proof superposition with dervation and explains

1.	Proof superposition with dervation and explains
Answer» Answer: Linear differential equations can always be written as: Lu = f, where L is a linear differential operator (example: 1st or 2ND derivative operator), u is the unknown function of interest, and f is some inhomogeneous term, which is a function of the time/space variables (for HOMOGENEOUS equations, set f = 0). By the DEFINITION of linear operator, for any two functions f, g and constants a, b, we have: L(af + bg) = aL(f) + bL(g). Thus, if u is a solution of the inhomogeneous equation, i.e. Lu = f, and if v is a solution of the homogeneous equation, i.e. Lv = 0, then we have: L(u + v) = Lu + Lv = f + 0 = f. We CONCLUDE that (u + v) is also a solution of the

Answer»

Answer:

Linear differential equations can always be written as:

Lu = f,

where L is a linear differential operator (example: 1st or 2ND derivative operator), u is the unknown function of interest, and f is some inhomogeneous term, which is a function of the time/space variables (for HOMOGENEOUS equations, set f = 0).

By the DEFINITION of linear operator, for any two functions f, g and constants a, b, we have:

L(af + bg) = aL(f) + bL(g).

Thus, if u is a solution of the inhomogeneous equation, i.e. Lu = f, and if v is a solution of the homogeneous equation, i.e. Lv = 0, then we have:

L(u + v) = Lu + Lv = f + 0 = f.

We CONCLUDE that (u + v) is also a solution of the

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