Question

Section 5.2 Problem 13:

Answer 1

The characteristic equation is

$r^2 + 14r + 50 = 0$

with complex roots r = -7 ± i, so the characteristic solution is

$y_c = C_1 \cos(7x) + C_2 \sin(7x)$

whose derivative is

${y_c}' = -7C_1 \sin(7x) + 7C_2 \cos(7x)$

Use the initial conditions to solve for the constants:

$y(0) = 2 \implies 2 = C_1$

$y'(0) = -17 \implies -17 = 7C_2 \implies C_2 = -\dfrac{17}7$

Then the particular solution is

$\boxed{y(x) = 2 \cos(7x) - \dfrac{17}7 \sin(7x)}$