Question

Find the general solution of the given differential equation y"p + y = 3 sin 2t + t cos 2t

Answer 1

The general solution to the non-homogeneous equation is then obtained by summing the particular solution
`\(y_p(t)\)` with the general solution of the homogeneous equation
`\(y(t) = C_1 \cos(2t) + C_2 \sin(2t) + (t^2)/(2) \cos(2t) - (3)/(10) \sin(2t)\)`

Step-by-step explanation:

The given second-order linear homogeneous differential equation
`\(y'' + y = 3 \sin(2t) + t \cos(2t)\)`is solved by employing the method of undetermined coefficients. First, the general solution of the associated homogeneous equation
`\(y'' + y = 0\)` is found to be
`\(y_h(t) = C_1 \cos(t) + C_2 \sin(t)\), where \(C_1\) and \(C_2\)`are arbitrary constants.

To solve for the particular solution, the right-hand side of the non-homogeneous equation
`\(3 \sin(2t) + t \cos(2t)\)` is broken down into two parts:
`\(3 \sin(2t)\) and \(t \cos(2t)\).` Assuming a particular solution in the form
`\(y_p(t) = At^2 \cos(2t) + Bt^2 \sin(2t)\)`is substituted into the differential equation.

By equating coefficients and solving the resulting system of equations, the particular solution is found to be
`\(y_p(t) = (t^2)/(2) \cos(2t) - (3)/(10) \sin(2t)\).`

The general solution to the non-homogeneous equation is then obtained by summing the particular solution
`\(y_p(t)\)` with the general solution of the homogeneous equation
`\(y_h(t)\), yielding \(y(t) = C_1 \cos(2t) + C_2 \sin(2t) + (t^2)/(2) \cos(2t) - (3)/(10) \sin(2t)\).`