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MAT 2324, Winter 2022, Quiz 4, Solutions

Q1: Consider the inhomogeneous equation

y'' - 2y' + 5y = g(x)

According to the list for how to choose possible particular solutions: If g is an exponential, we choose Y to be an exponential etc. The only caveat to this list is that if g or parts of it are already a solution to the homogeneous problem, then we cannot use those for Y but have to instead multiply them with the independent variable first.

Hence, we calculate the solutions of the homogeneous equation. Using the characteristic equation, we find

Aex cos(2x) + Bex sin(2x)

Hence, this guess for Y must be excluded from the list.

Q2: (T/F) The method of undetermined coefficients can be applied to all constant coefficient equations.

The statement is false. What is correct is that the method of undetermined coefficients can ONLY be applied to constant coefficient equations, but not to ALL.

If the inhomogeneous term is not one on the list for possible Y, we cannot apply the method, even if the equation has constant coefficients. There is an example in the textbook.

Q3: (T/F) The variation of constants formula for Euler-Cauchy equations

The formula can be applied to Euler-Cauchy equations, but it requires that the equation is in standard form, i.e., that the coefficient in front of the highest derivative (the second) is unity. If that is not the case, we need to divide by that coefficient. Dividing by the coefficient will change the function g(t) on the right-hand side. The formula needs to reflect this.

Again, there is an example in the textbook.