Differential equation help please?

Question

The governing equation of system if given by: d^2y/dt^2 + 4y = cos 2t

1.) I need to find the complementary function

2.) given the particular integral of this equation is y = 1/4t sin 2t, find the solution which satifies the boundary condition y(0)=dy/dt (0) = 0

Thanks in advance.

Answer 1

The "complementary function" is just another name for the solution to the homogeneous version of the differential equation, in this case the homogeneous equation is:

y'' + 4y = 0

Using the usual trial solution y = exp(-k*t), we get the characteristic equation:

k^2 + 4 = 0

which has roots:

k = +2i, -2i

In this case, the roots are imaginary, so the homogeneous solution (the complementary function) y_h(t) is:

y_h(t) = A*sin(2*t) + B*cos(2*t)

where A and B are constants of integration that need to be determined from specific boundary conditions.

You are given the particular solution to the original differential equation.  Note that this can easily be found by using the method of undetermined coefficients.

The complete solution to the original equation is given by the sum of the homogeneous and particular solutions:

y(t) = A*sin(2*t) + B*cos(2*t) + (t/4)*sin(2*t)

Now find A and B by using the boundary conditions you are given.  To do this, we need to first obtain the expression for y':

y' = 2*A*cos(2*t) - 2*B*sin(2*t) + (t/2)*cos(2*t) + (1/4)*sin(2*t)

Now set y(0) = y'(0) = 0 and solve for A and B:

y(0) = 0 = A*sin(0) + B*cos(0) + (0)*sin(0)

0 = B

y'(0) = 0 = 2*A*cos(0) - 2*B*sin(0) + (0)*cos(0) + (1/4)*sin(0)

0 = A

So in this case, the solution that satisfies the given boundary conditions is simply the particular solution:

y(t) = (t/4)*sin(2*t)