Question

State the principle of superposition and prove that it holds only for linear differential equations.​

Answer 1

Answer:

Explanation:

Well that’s easy to show.

Let Y and Z be unique, non-trivial solutions to a differential equation y^2+y’+y’’=0. Consider then H=Y+Z

Plug H=Y+Z into the differential equation and you get

Z^2+2ZY+Y^2+Z’+Y’+Z’’+Y’’

Here you’ll notice Y^2+Y’+Y’’ and Z^2+Z’+Z’’ are zero.

Which means we’re left with 2YZ which may or may not be zero. In fact, it’s only zero if either Z or Y are zero. Which obviously violates our nontrivial assumption at the start.

So it must be the case 2YZ is non zero, and H doesn’t satisfy the differential equation.

This means your conjecture here that it ought to apply to nonlinear equations is disproven by counter example.

More generally, superposition relies on the fact that the differential operator is a linear operator.

Linear Operator -- from Wolfram MathWorld

Answer 2

Answer:

hahahaha that's the only thing is that