# Differential Equations : Undetermined Coefficients

## Example Questions

### Example Question #4 : Higher Order Differential Equations

Solve the given differential equation by undetermined coefficients.

Explanation:

First solve the homogeneous portion:

Therefore,  is a repeated root thus one of the complimentary solutions  is,

Now find the remaining complimentary solution .

Now solve for  and .

Where

and

Therefore,

Now, combine both of the complimentary solutions together to arrive at the general solution.

### Example Question #1 : Undetermined Coefficients

Find the general solution for .

Explanation:

This is a higher order inhomogeneous linear differential equation. Because the inhomogeneity is a cosine, we will use variation of parameters to solve it.

First, we find the characteristic equation to solve for the homogenous solution. This gives us .

This tells us that the homogeneous solution is . As neither of these overlap with our inhomogeneity, we are safe to continue without adding a factor of t.

Thus, let us guess that . Then,

and

Plugging into the original equation, we have

Which implies that  and . Solving via substitution,

Thus, the particular solution is  and the overall solution is the particular plus the homogeneous.

So

### Example Question #2 : Undetermined Coefficients

Find the form of a particular solution to the following differential equation that could be used in the method of undetermined coefficients:

The form of a particular solution is    where A and B are real numbers.

The form of a particular solution is    where A and B are real numbers.

The form of a particular solution is    where A,B, and C are real numbers.

The form of a particular solution is    where A,B, C, and D are real numbers.

The form of a particular solution is    where A,B, and C are real numbers.

Explanation:

We first note that the differential equation has characteristic equation

,

since the roots of this characteristic polynomial are linearly independent of the forcing function

,

we simply use undetermined coefficient combination rules to figure that the particular solution will be of form

### Example Question #3 : Undetermined Coefficients

Consider the differential equation

The particular solution used in undetermined coefficients will be of what form?

The particular solution will be of form:

where A,B,C,D,E, and F are real numbers

The particular solution will be of form:

where A,B, and C are real numbers

The particular solution will be of form:

where A and B are real numbers

The particular solution will be of form:

where A is a real number

The particular solution will be of form:

where A,B,C,D,E, and F are real numbers

Explanation:

We first figure that the forcing function   is linearly independent  to the homogeneous solution solved with the characteristic equation.

Therefore, using proper undetermined coefficients function rules, the particular solution will be of the form:

It is important to note that when either a sine or a cosine is used, both sine and cosine must show up in the particular solution guess.

### Example Question #4 : Undetermined Coefficients

Solve for a particular solution of the differential equation using the method of undetermined coefficients.

Explanation:

We start with the assumption that the particular solution must be of the form

.

Then we solve the first and second derivatives with this assumption, that is,

and .

Then we plug in these quantities into the given equation to get:

, which solves for .

Thus, but the method of undetermined coefficients, a particular solution to this differential equation is:

### Example Question #5 : Undetermined Coefficients

Solve the given differential equation by undetermined coefficients.

Explanation:

First solve the homogeneous portion:

Therefore,  is a repeated root thus one of the complimentary solutions  is,

Now find the remaining complimentary solution .

Now solve for  and .

Where

and

Therefore,

Now, combine both of the complimentary solutions together to arrive at the general solution.

### Example Question #6 : Undetermined Coefficients

Find the form of a particular solution to the following Differential Equation (Do NOT Solve)