- #1

arhzz

- 268

- 52

- Homework Statement
- Solve the differential equation using the fundemental matrix and derivation of the constant

- Relevant Equations
- -

I am given this system of differential equations;

$$ x_1'=2t^2x_1+3t^2x_2+t^5 $$

$$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$

Now the first question states the following;

Find a fundamental matrix of the corresponding homogeneous system and

explain exactly how you arrive at independent solutions

And the second question;

We consider the DGL system from Exercise 15. Find all the solutions

with variation of the constants.

Now really it is just like having a part a) and b) of question.To solve the first question (part a if you will) I did this;

So the corresponding homogenous system should be

$$ x_1'=2t^2x_1+3t^2x_2 $$

$$ x_2' =-2t^2x_1-3t^2x_2 $$

Than I created the coefficent matrix and find the eigenvalues and eigenvectors;\begin {matrix}

t^2*(2 & 3) \\

(-2 & -3)

\end{matrix}

Note: the ##t^2 ## should be infront of the matrix multiplying the whole thing,but not sure how to do it in LaTeX properly;

I will for the sake of the length of the post skip how I got the eigenvalues but they should be ## \lambda_1 = 0 ## and ##\lambda_2 = -t^2 ##

Now I found the eigenvectors using wolfram alpha and they should be ## v_1 = (1,\frac{-2}{3})^T ## and ## v_2 =(1,-1)^T ## So these are the vectors that I need for my fundamental matrix.Now I have to create the fundamental matrix.I did that like this;

$$ y_1 = (v1,v2) * e^{\lambda_1*t} $$ and analog with ##\lambda_2 ## and the eigenvectors of that eigenvalue

My fundamental matrix looks like this

\begin {matrix}

1 & e^{-t}\\

\frac{-2}{3} & -e^{-t})

\end{matrix}

Now I thought this was right,and the tutor for the class pointed out this; "the constant t should also be included". I am not really sure what that means and I am pretty sure that I cannot answer the second question if this one is not right. Do I simply need to multiply this matrix with t?

Thanks!

$$ x_1'=2t^2x_1+3t^2x_2+t^5 $$

$$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$

Now the first question states the following;

Find a fundamental matrix of the corresponding homogeneous system and

explain exactly how you arrive at independent solutions

And the second question;

We consider the DGL system from Exercise 15. Find all the solutions

with variation of the constants.

Now really it is just like having a part a) and b) of question.To solve the first question (part a if you will) I did this;

So the corresponding homogenous system should be

$$ x_1'=2t^2x_1+3t^2x_2 $$

$$ x_2' =-2t^2x_1-3t^2x_2 $$

Than I created the coefficent matrix and find the eigenvalues and eigenvectors;\begin {matrix}

t^2*(2 & 3) \\

(-2 & -3)

\end{matrix}

Note: the ##t^2 ## should be infront of the matrix multiplying the whole thing,but not sure how to do it in LaTeX properly;

I will for the sake of the length of the post skip how I got the eigenvalues but they should be ## \lambda_1 = 0 ## and ##\lambda_2 = -t^2 ##

Now I found the eigenvectors using wolfram alpha and they should be ## v_1 = (1,\frac{-2}{3})^T ## and ## v_2 =(1,-1)^T ## So these are the vectors that I need for my fundamental matrix.Now I have to create the fundamental matrix.I did that like this;

$$ y_1 = (v1,v2) * e^{\lambda_1*t} $$ and analog with ##\lambda_2 ## and the eigenvectors of that eigenvalue

My fundamental matrix looks like this

\begin {matrix}

1 & e^{-t}\\

\frac{-2}{3} & -e^{-t})

\end{matrix}

Now I thought this was right,and the tutor for the class pointed out this; "the constant t should also be included". I am not really sure what that means and I am pretty sure that I cannot answer the second question if this one is not right. Do I simply need to multiply this matrix with t?

Thanks!