Question

Solve the differential equation using the method of undetermined coefficients.

y'' − 10y' + 26y = e^−x

Answer 1

Find the complementary solution by solving ( d^2 y(x))/( dx^2) - 10 ( dy(x))/( dx) + 26 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)) - 10 d/( dx)(e^(λ x)) + 26 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) - 10 λ e^(λ x) + 26 e^(λ x) = 0

Factor out e^(λ x):

(λ^2 - 10 λ + 26) e^(λ x) = 0

Since e^(λ x) !=0 for any finite λ, the zeros must come from the polynomial:

λ^2 - 10 λ + 26 = 0

Solve for λ:

λ = 5 + i or λ = 5 - i

The roots λ = 5 ± i give y_1(x) = c_1 e^((5 + i) x), y_2(x) = c_2 e^((5 - i) 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^((5 + i) x) + c_2 e^((5 - i) x)

Apply Euler's identity e^(α + i β) = e^α cos(β) + i e^α sin(β):

y(x) = c_1 (e^(5 x) cos(x) + i e^(5 x) sin(x)) + c_2 (e^(5 x) cos(x) - i e^(5 x) sin(x))

Regroup terms:

y(x) = (c_1 + c_2) e^(5 x) cos(x) + i (c_1 - c_2) e^(5 x) sin(x)

Redefine c_1 + c_2 as c_1 and i (c_1 - c_2) as c_2, since these are arbitrary constants:

y(x) = c_1 e^(5 x) cos(x) + c_2 e^(5 x) sin(x)

Determine the particular solution to ( d^2 y(x))/( dx^2) - 10 ( dy(x))/( dx) + 26 y(x) = e^(-x) by the method of undetermined coefficients:

The particular solution to ( d^2 y(x))/( dx^2) - 10 ( dy(x))/( dx) + 26 y(x) = e^(-x) is of the form:

y_p(x) = a_1 e^(-x)

Solve for the unknown constant a_1:

Compute ( dy_p(x))/( dx):

( dy_p(x))/( dx) = d/( dx)(a_1 e^(-x))

= -a_1 e^(-x)

Compute ( d^2 y_p(x))/( dx^2):

( d^2 y_p(x))/( dx^2) = ( d^2 )/( dx^2)(a_1 e^(-x))

= a_1 e^(-x)

Substitute the particular solution y_p(x) into the differential equation:

( d^2 y_p(x))/( dx^2) - 10 ( dy_p(x))/( dx) + 26 y_p(x) = e^(-x)

a_1 e^(-x) - 10 (-a_1 e^(-x)) + 26 (a_1 e^(-x)) = e^(-x)

Simplify:

37 a_1 e^(-x) = e^(-x)

Equate the coefficients of e^(-x) on both sides of the equation:

37 a_1 = 1

Solve the equation:

a_1 = 1/37

Substitute a_1 into y_p(x) = a_1 e^(-x):

y_p(x) = e^(-x)/37

The general solution is:

Answer: |

| y(x) = y_c(x) + y_p(x) = e^(-x)/37 + c_1 e^(5 x) cos(x) + c_2 e^(5 x) sin(x)