Question

1252) y=(C1)exp(Ax)+(C2)exp(Bx)+F+Gx is the general solution of
the second order linear differential equation:
   (y'') + ( 6y') + (-27y) = ( 2) + ( -3)x. Find A,B,F,G, where
A>B. This exercise may show "+ (-#)" which should be enterered into
the calculator as "-#", and not "+-#". ans:4

Answer 1

Find the complementary solution by solving ( d^2 y(x))/( dx^2) + 6 ( dy(x))/( dx) - 27 y(x) = 0:
Assume a solution will be proportional to e^(λ x) for some constant λ.
Substitute y(x) = e^(λ x) into the differential equation:
( d^2 )/( dx^2)(e^(λ x)) + 6 d/( dx)(e^(λ x)) - 27 e^(λ x) = 0
Substitute ( d^2 )/( dx^2)(e^(λ x)) = λ^2 e^(λ x) and d/( dx)(e^(λ x)) = λ e^(λ x):
λ^2 e^(λ x) + 6 λ e^(λ x) - 27 e^(λ x) = 0
Factor out e^(λ x):
(λ^2 + 6 λ - 27) e^(λ x) = 0
Since e^(λ x) !=0 for any finite λ, the zeros must come from the polynomial:
λ^2 + 6 λ - 27 = 0
Factor:
(λ - 3) (λ + 9) = 0
Solve for λ:
λ = -9 or λ = 3
The root λ = -9 gives y_1(x) = c_1 e^(-9 x) as a solution, where c_1 is an arbitrary constant.
The root λ = 3 gives y_2(x) = c_2 e^(3 x) as a solution, where c_2 is an arbitrary constant.
The general solution is the sum of the above solutions:
y(x) = y_1(x) + y_2(x) = c_1 e^(-9 x) + c_2 e^(3 x)
Determine the particular solution to ( d^2 y(x))/( dx^2) + 6 ( dy(x))/( dx) - 27 y(x) = -3 x + 2 by the method of undetermined coefficients:
The particular solution to ( d^2 y(x))/( dx^2) + 6 ( dy(x))/( dx) - 27 y(x) = -3 x + 2 is of the form:
y_p(x) = a_1 + a_2 x
Solve for the unknown constants a_1 and a_2:
Compute ( dy_p(x))/( dx):
( dy_p(x))/( dx) = d/( dx)(a_1 + a_2 x)
= a_2
Compute ( d^2 y_p(x))/( dx^2):
( d^2 y_p(x))/( dx^2) = ( d^2 )/( dx^2)(a_1 + a_2 x)
= 0
Substitute the particular solution y_p(x) into the differential equation:
( d^2 y_p(x))/( dx^2) + 6 ( dy_p(x))/( dx) - 27 y_p(x) = -3 x + 2
6 a_2 - 27 (a_1 + a_2 x) = -3 x + 2
Simplify:
-27 a_1 + 6 a_2 - 27 a_2 x = 2 - 3 x
Equate the coefficients of 1 on both sides of the equation:
-27 a_1 + 6 a_2 = 2
Equate the coefficients of x on both sides of the equation:
-27 a_2 = -3
Solve the system:
a_1 = -4/81
a_2 = 1/9
Substitute a_1 and a_2 into y_p(x) = a_1 + x a_2:
y_p(x) = x/9 - 4/81
The general solution is:
Answer: |
| y(x) = y_c(x) + y_p(x) = x/9 + c_1 e^(-9 x) + c_2 e^(3 x) - 4/81