2. Which of the following P.D.E.’s is linear? Justify your answer. (a) 2ut = xtuxx + etux + u.
(b) 2ut = xuuxx + etux + t.

Question

amistre64 · Answer

id say a, but i dont have a good reason as to why, or even if its the right choice

anonymous · Answer

I'm checking using L(u1+u2) = L(u1) + L(u2) and L(cU) = cL(u).  Not sure if this is the right approach for 2nd order pde..

amistre64 · Answer

me either, you prolly know ten times as much as me already about it tho

anonymous · Answer

lol.

anonymous · Answer

lol.

jamesj · Answer

Yes, you're right: those are exactly the linearity conditions.  So run 'em through and see.  You can also do them both at the same time by testing to see whether 

P(c1.u1 + c2.u2) = c1.P(u1) + c2.P(u2)

Where P is the differential operator of the PDE and c1 and c2 are arbitrary constants.

jamesj · Answer

I.e., if P satisfies this condition, then indeed P is a linear operator and the associated PDE is a linear PDE.