Question

Find the constants "a" and "b" if y = e^(4x)(2x + 1) is a solution of the differential equation d²y/dx² - a(dy/dx) + by = 0 by the product rule.

a. a = 2, b = 8
b. a = 4, b = 16
c. a = 6, b = 24
d. a = 8, b = 32

Answer 1

To find the constants 'a' and 'b', we need to substitute the given solution 'y = e^(4x)(2x + 1)' into the differential equation and equate coefficients. By comparing the coefficients of e^(4x) and the constant terms, we can solve for 'a' and 'b'. The correct answer is a = 4 and b = 16.

Step-by-step explanation:

To find the constants a and b, we need to substitute the given solution y = e^(4x)(2x + 1) into the differential equation. Using the product rule, we differentiate y twice with respect to x, which gives us:

dy/dx = 4e^(4x)(2x + 1) + e^(4x)(2)

d^2y/dx^2 = 16e^(4x)(2x + 1) + 4e^(4x)

Substituting these derivatives and the given solution into the differential equation d^2y/dx^2 - a(dy/dx) + by = 0, we get:

(16e^(4x)(2x + 1) + 4e^(4x)) - a(4e^(4x)(2x + 1) + e^(4x)(2)) + be^(4x)(2x + 1) = 0

Expanding and simplifying this equation, we can equate the coefficients of e^(4x) and the constant terms to find a and b. Comparing the coefficients, we get:

16 - 4a + 2b = 0 (for the coefficient of e^(4x))

4 - a = 0 (for the constant term)

Solving these linear equations simultaneously, we find that a = 4 and b = 16. Therefore, the correct answer is option (b) a = 4, b = 16.