# What is a solution to the differential equation #y'' + 4y = 8sin2t#?

##### 1 Answer

The General Solution to the DE

#### Explanation:

There are two major steps to solving Second Order DE's of this form:

**Find the Complementary Function (CF)**

This means find the general solution of the Homogeneous Equation

To do this we look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

As

This has two distinct complex solutions,

And so the solution to the DE is;

-+-+-+-+-+-+-+-+-+-+-+-+-+-+

Verification:

If

And so,

-+-+-+-+-+-+-+-+-+-+-+-+-+-+

**Find a Particular Integral* (PI)**

This means we need to find a specific solution (that is not already part of the solution to the Homogeneous Equation).

We would normally look for a particular solution which is a combination of functions on the RHS of the form

However these are both solutions of the homogeneous system, so instead we must look for solution of the form

If

Differentiating again, we get:

If we substitute into the DE

Equating Coefficients of

So we have found that a Particular Solution is:

ie

**General Solution (GS)**

The General Solution to the DE is then:

GS = CF + PI

Hence The General Solution to the DE