Question

Consider the following differential equation to be solved by the method of undetermined coefficients. \[ y^{\prime \prime}-2 y^{\prime}+50 y=e^{x} \cos (7 x) \] Find the complementary function for the

Answer 1

The complementary function for the given differential equation is
`\(y_c = C_1e^(x)\cos(7x) + C_2e^(x)\sin(7x)\).`

Step-by-step explanation:

To find the complementary function, we first consider the homogeneous part of the differential equation, setting the right-hand side to zero. The homogeneous equation is
`\(y'' - 2y' + 50y = 0\)`. To solve this, we assume a solution of the form
`\(y_c = e^(rx)\), where \(r\)` is a constant. Substituting this into the homogeneous equation, we get the characteristic equation
`\(r^2 - 2r + 50 = 0\)`. Solving this quadratic equation yields complex roots
`\(r_1 = 1 + 7i\) and \(r_2 = 1 - 7i\)`.

The general solution for the homogeneous part is then
`\(y_c = C_1e^((1+7i)x) + C_2e^((1-7i)x)\)`. Using Euler's formula
`(\(e^(ix) = \cos(x) + i\sin(x)\))`, we can rewrite this a
`s \(y_c = C_1e^(x)\cos(7x) + C_2e^(x)\sin(7x)\)`.

This is the complementary function, representing the general solution of the homogeneous differential equation. The constants
`\(C_1\) and \(C_2\)` are determined by initial conditions if provided in the specific problem. The particular solution, which accounts for the non-homogeneous term
`\(e^(x)\cos(7x)\)`, can be found separately and added to the complementary function to obtain the complete solution of the given differential equation.