Question

(1 point) Solve the following differential equation by variation of parameters. Fully evaluate all integrals. y" +9y sec(3x) a. Find the most general solution to the associated homogeneous differential equation. Use c1 and c2 in your answer to denote arbitrary constants, and enter them as ct and c2. help (formulas) b. Find a particular solution to the nonhomogeneous differential equation y" +9y sec(3x). yp elp (formulaS c. Find the most general solution to the original nonhomogeneous differential equation. Use c and c2 in your answer to denote arbitrary constants help (formulas)

Answer 1

"Given differential equation is y"" + 9y = sec (3x) ------(1) we have to solve it by using variation of parameters Now consider the homogeneous part of (1) i.e y"" + 9y = 0 rightarrow (2) or (D^2 + 9)y = 0: where D = d/dx its auxiliary equation is m^2 + 9 = 0, so it gives m = plusminus i3 and hence the solution of (2) i.e the solution of your part (a) is given by (a) y_n = c_1 cos 3x + c_2 sin 3x where c_1 and c_2 are arbitrary constants Now let y = A cos 3x + B sin 3x rightarrow (3) be the complete solution of (1) then A and B are function of x which are so choose that (1) will be Differentiate equation (3) w.r.to x we have y' = A' cos 3x - A 3 sin 3x + B' sin 3x + B 3 cos x ------(4)"