The homogeneous equation
has non-constant, non-singular coefficients and . Two solutions of the equation, and , are given. The solutions are known to be such that the determinant
is non-zero for all . Define what is meant by linear dependence, and show that the two given solutions are linearly independent. Show also that
In the corresponding inhomogeneous equation
the right-hand side is a prescribed forcing function. Construct a particular integral of this inhomogeneous equation in the form
where the two functions are to be determined such that
for all . Express your result for the functions in terms of integrals of the functions and .
Consider the case in which for all and is a positive constant, say, and in which the forcing . Show that in this case and can be taken as and respectively. Evaluate and and show that, as , one of the increases in magnitude like a power of to be determined.