Question 3:
You find that the general solution is
v = y/x = Ae^2x + Be^-2x - e^x
But when you multiply both sides by x, this becomes
y = Axe^2x + Bxe^-2x - e^x
In which e^x is not multiplied by x, and is also a particular integral.
Similarly, in question 5:
You find that the general solution is
u = xy = Ae^-3x + Bxe^-3x + 3x - 2
However, this time when both sides are divided by x, you get
y = A/x*e^-3x + Be^-3x + 3 - 2/x
In which the particular integral, 3x - 2, has been changed.
My question is, do all particular integrals remain unchanged during substitution? If so, why? If not, why not?
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