Question

Problem 8 (14 points). Using the method of undetermined coefficients, find a particular solution Yp of the equation y" - Sy' +16y = 4x +2. Then find the general solution of this equation.

Answer 1

Find the complementary solution by solving ( d^2 y(x))/( dx^2) - 8 ( dy(x))/( dx) + 16 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)) - 8 d/( dx)(e^(λ x)) + 16 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) - 8 λ e^(λ x) + 16 e^(λ x) = 0
Factor out e^(λ x):
(λ^2 - 8 λ + 16) e^(λ x) = 0
Since e^(λ x) !=0 for any finite λ, the zeros must come from the polynomial:
λ^2 - 8 λ + 16 = 0
Factor:
(λ - 4)^2 = 0
Solve for λ:
λ = 4 or λ = 4
The multiplicity of the root λ = 4 is 2 which gives y_1(x) = c_1 e^(4 x), y_2(x) = c_2 e^(4 x) x as solutions, where c_1 and c_2 are arbitrary constants.
The general solution is the sum of the above solutions:
y(x) = y_1(x) + y_2(x) = c_1 e^(4 x) + c_2 e^(4 x) x
Determine the particular solution to ( d^2 y(x))/( dx^2) - 8 ( dy(x))/( dx) + 16 y(x) = 4 x + 2 by the method of undetermined coefficients:
The particular solution to ( d^2 y(x))/( dx^2) - 8 ( dy(x))/( dx) + 16 y(x) = 4 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) - 8 ( dy_p(x))/( dx) + 16 y_p(x) = 4 x + 2
-8 a_2 + 16 (a_1 + a_2 x) = 4 x + 2
Simplify:
16 a_1 - 8 a_2 + 16 a_2 x = 2 + 4 x
Equate the coefficients of 1 on both sides of the equation:
16 a_1 - 8 a_2 = 2
Equate the coefficients of x on both sides of the equation:
16 a_2 = 4
Solve the system:
a_1 = 1/4
a_2 = 1/4
Substitute a_1 and a_2 into y_p(x) = a_1 + x a_2:
y_p(x) = x/4 + 1/4
The general solution is:
Answer: |
| y(x) = y_c(x) + y_p(x) = x/4 + c_1 e^(4 x) + c_2 e^(4 x) x + 1/4