Homogeneous Linear Systems - Differential Equations

To solve the homogeneous system, we will need a fundamental matrix. Specifically, it will help to get the matrix exponential. To do this, we will diagonalize the matrix. First, we will find the eigenvalues which we can do by calculating the determinant of .

Finding the eigenspaces, for lambda = 1, we have

Adding -1/2 Row 1 to Row 2 and dividing by -1/2, we have which means

Thus, we have an eigenvector of .

For lambda = 4

Adding Row 1 to Row 2, we have

So with an eigenvector .

Thus, we have and . Using the inverse formula for 2x2 matrices, we have that . As we know that , we have

The solution to a homogenous system of linear equations is simply to multiply the matrix exponential by the intial condition. For other fundamental matrices, the matrix inverse is needed as well.

Thus, our final answer is

So this is a homogenous, second order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:

Which, using the quadratic formula or factoring gives us roots of and

The solution of homogenous equations is written in the form:

so we don't know the constants, but can substitute the values we solved for the roots:

We have two initial values, one for y(t) and one for y'(t), both with t=0\

So:

so:

We can solve for : Then plug into the other equation to solve for

So, solving, we get: Then

This gives a final answer of:

So this is a homogenous, third order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:

Which, using the cubic formula or factoring gives us roots of , and

The solution of homogenous equations is written in the form:

so we don't know the constants, but can substitute the values we solved for the roots:

We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0

So:

so:

So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: Then and

This gives a final answer of:

So this is a homogenous, second order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:

Which, using the quadratic formula or factoring gives us roots of and

The solution of homogenous equations is written in the form:

so we don't know the constants, but can substitute the values we solved for the roots:

We have two initial values, one for y(t) and one for y'(t), both with t=0

So:

so:

We can solve Then plug into the other equation to solve for

So, solving, we get: Then

This gives a final answer of:

So this is a homogenous, first order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:

gives us a root of

The solution of homogenous equations is written in the form:

so we don't know the constant, but can substitute the values we solved for the root:

We have one initial values, for y(t) with t=0

So:

This gives a final answer of:

So this is a homogenous, third order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:

Which, using the cubic formula or factoring gives us roots of , and

The solution of homogenous equations is written in the form:

so we don't know the constants, but can substitute the values we solved for the roots:

We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0

So:

So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: Then and

This gives a final answer of:

To find the general solution to the given system

first find the eigenvalues and eigenvectors.

Therefore the eigenvalues are

Now calculate the eigenvectors

For

Thus,

For

Thus

Therefore,

Now the general solution is,

A saddle point phase plane results from two real eigenvalues of different signs. Three of these matrices are triangular, which means their eigenvalues are on the diagonal. For these three, the eigenvalues are real, but both the same sign, meaning they don't have saddles. For the remaining two, we'll need to find the eigenvalues using the characteristic equations.

For , we have

The discriminant to this is , so the solutions are non-real. Thus, this matrix doesn't yield a saddle point.

For we have,

We see that this matrix yields two real eigenvalues with different signs. Thus, it is the correct choice.

Finding the eigenvalues and eigenvectors of with the characteristic equation of the matrix

The corresponding eigenvalues are, respectively

and

This gives us that the general solution is