The characteristic equation is
[tex]r^2 + 14r + 50 = 0[/tex]
with complex roots r = -7 ± i, so the characteristic solution is
[tex]y_c = C_1 \cos(7x) + C_2 \sin(7x)[/tex]
whose derivative is
[tex]{y_c}' = -7C_1 \sin(7x) + 7C_2 \cos(7x)[/tex]
Use the initial conditions to solve for the constants:
[tex]y(0) = 2 \implies 2 = C_1[/tex]
[tex]y'(0) = -17 \implies -17 = 7C_2 \implies C_2 = -\dfrac{17}7[/tex]
Then the particular solution is
[tex]\boxed{y(x) = 2 \cos(7x) - \dfrac{17}7 \sin(7x)}[/tex]
