What is the particular solution of the differential equation # y''' + 3y''-4y'-12y =0 #?
The solution is # y = Ae^(-2x) + Be^(2x) + Ce^(-3x)#
The solution is
1 Answer
We have:
# y''' + 3y''-4y'-12y =0 # ..... [A]
This is a Third order linear Homogeneous Differentiation Equation with constant coefficients. The standard approach is to find a solution,
Complimentary Function
The Auxiliary equation associated with the homogeneous equation of [A] is:
# m^3 +3m^2-4m-12 = 0 #
The challenge with higher order Differential Equation is solving the associated higher order Axillary equation. By inspection we see
# (m-2)(m^2+5m+6) = 0#
# :. (m-2)(m+2)(m+3) = 0#
Which has three real distinct solution
The roots of the axillary equation determine parts of the solution, which if linearly independent then the superposition of the solutions form the full general solution.
- Real distinct roots
#m=alpha,beta, ...# will yield linearly independent solutions of the form#y_1=Ae^(alphax)# ,#y_2=Be^(betax)# , ... - Real repeated roots
#m=alpha# , will yield a solution of the form#y=(Ax+B)e^(alphax)# where the polynomial has the same degree as the repeat. - Complex roots (which must occur as conjugate pairs)
#m=p+-qi# will yield a pairs linearly independent solutions of the form#y=e^(px)(Acos(qx)+Bsin(qx))#
Thus the solution of the homogeneous equation is:
# y = Ae^(-2x) + Be^(2x) + Ce^(-3x)#
Note this solution has
This is consistent with the suggested solution.