# Find the general solution to the differential equation: y ” – 2y ‘+ 5y = cos (7x).

Consider the differential equation y ” – 2 * y ‘+ 5 * y = cos (7 * x). Analysis of this equation shows that it is a linear inhomogeneous differential equation of the second order with constant coefficients. First, we compose and solve the corresponding linear homogeneous equation y ” – 2 * y ‘+ 5 * y = 0.

The characteristic equation for the last equation is: k² – 2 * k + 5 = 0. Find the discriminant of the quadratic equation: D = b2– 4 * a * c = (-2) 2 – 4 * 1 * 5 = 4 – 20 = – 16. Since the discriminant is less than zero, the quadratic equation has no real solutions. Its complex solutions will be: k1 = 1 – 2 * i and k1 = 1 + 2 * i. Therefore, the solution to the homogeneous equation is: y = ex * (C1 * cos (2 * x) + C2 * sin (2 * x)), where C1 and C2 are constants.

Consider the right-hand side of this inhomogeneous equation, which we denote by f (x). It has the form: f (x) = cos (7 * x). Therefore, we are looking for a particular solution of the inhomogeneous equation in the form y = A * cos (7 * x) + B * sin (7 * x). We calculate the derivatives: yꞋ = (A * cos (7 * x) + B * sin (7 * x)) Ꞌ = -7 * A * sin (7 * x) + 7 * B * cos (7 * x) and yꞋꞋ = (yꞋ) Ꞌ = (-7 * A * sin (7 * x) + 7 * B * cos (7 * x)) Ꞌ = -49 * (A * cos (7 * x) + B * sin (7 * x)).

Substitute the found expressions into the original differential equation. Then, we get: y ” – 2 * y ‘+ 5 * y = -49 * (A * cos (7 * x) + B * sin (7 * x)) – 2 * (-7 * A * sin ( 7 * x) + 7 * B * cos (7 * x)) + 5 * (A * cos (7 * x) + B * sin (7 * x)) = cos (7 * x) or 14 * A * sin (7 * x) – 44 * A * cos (7 * x) – 44 * B * sin (7 * x) – 14 * B * cos (7 * x) = cos (7 * x).

Equating the coefficients for the same functions, we obtain the following system of equations: (with sin (7 * x)) the equation 14 * A – 44 * B = 0 and (with cos (7 * x)) the equation -44 * A – 14 * B = 1. Solving it, we find: A = -11/533; B = -7/1066. So, a particular solution looks like: y = (-11/533) * cos (7 * x) – (7/1066) * sin (7 * x).

Thus, the general solution of this differential equation is: y = ex * (C1 * cos (2 * x) + C2 * sin (2 * x)) – (11/533) * cos (7 * x) – (7 / 1066) * sin (7 * x), where C1 and C2 are constants.