Question

find the general solution to the differential equation xy' + 18y = e^(x^18)

Answer 1

Use the principle of superposition, and you will find that the solution for any function that is in the form [y(t) = c1e^3t + c2e^-3t is a solution for the equation. So use a function that comes back to itself after 2 derivatives, an exponential function, and with the proper exponent e^xt , the 18 will be taken care of as well! You could use e^3t, 3e^3t, or 6e^3t, and even 9e^3t. This is assuming that this function you placed is homogenous, and g(t) = 0.

Answer 2

i..still dont get it..

Answer 3

Go back to your properties of Linear Homogenous Constant-Coefficient Second Order Differential Equations. You will find out how to verify a solution or derivative, to see that they are proved to be in fact what they are, and not just something made up.