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Thread: 6.1.1 Show that .... is a solution of.....

  1. #1
    Super Member bigwave's Avatar
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    6.1.1 Show that .... is a solution of.....

    $\textsf{6.1.1 Show that}\\$
    $$y=\begin{bmatrix}
    c_1e^{2x} + c_2e^{3x}\\
    2c_1e^{2x} + c_2e^{3x}
    \end{bmatrix}\\$$
    $\textsf{is a solution of} \\$
    $$Y'=\left[\begin{array}{rr}
    4 & -1 \\2 & 1
    \end{array}\right]Y\\$$
    $\textsf{6.1.2 Show that}\\$
    $$Y'=\left[\begin{array}{rrrr}
    c_1e^{2x}& + c_2e^{3x}& -\frac{x}{6}&-\frac{11}{36} \\ \\
    2c_1e^{2x}& + c_2e^{3x}& -\frac{2x}{6}&-\frac{1}{18}
    \end{array}\right]\\$$
    $\textsf{is a solution of} \\$
    $$Y'=\left[\begin{array}{rr}
    4 & -1 \\2 & 1
    \end{array}\right]Y
    +\left[\begin{array}{rr}
    1 \\x
    \end{array}\right]$$

    ok I realize these are simple but still the examples given confused me.
    I assume the first step is to span
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  2. #2
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    Re: 6.1.1 Show that .... is a solution of.....

    To show that Y is a solution to a given equation, put that Y into the equation!

    Here, $Y= \begin{bmatrix}c_1e^{2x}+ c_2e^{3x} \\ 2c_1e^{2x}+ c_2e^{3x}\end{bmatrix}$ and the equation is $\displaystyle Y'= \begin{bmatrix}4 & -1 \\ 2 & 1 \end{bmatrix}Y$

    Do you understand that $Y'= \begin{bmatrix}2c_1e^{2x}+ 3c_2e^{3x} \\ 4c_1e^{2x}+ 3c_2e^{3x}\end{bmatrix}$?

    What is $\begin{bmatrix}4 & -1 \\ 2 & 1 \end{bmatrix}\begin{bmatrix}c_1e^{2x}+ c_2e^{3x} \\ 2c_1e^{2x}+ c_2e^{3x}\end{bmatrix}$?
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  3. #3
    Super Member bigwave's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    $\begin{bmatrix}
    4&-1\\ 2&1
    \end{bmatrix}$
    $\begin{bmatrix}
    c_1e^{2x}&c_2e^{3x}\\
    2c_1e^{2x}&c_2e^{3x}
    \end{bmatrix}$
    $=\begin{bmatrix}
    2c_1e^{2x}&3c_2e^{3x}\\
    4c_1e^{2x}&3c_2e^{3x}
    \end{bmatrix}$
    $\begin{bmatrix}
    4c_1e^{2x}-2c_1e^{2x}&4c_2e^{3x}-c_2e^{3x}\\
    2c_1e^{2x}+2c_1e^{2x}&2c_2e^{3x}+c_2e^{3x}
    \end{bmatrix}$
    $=\begin{bmatrix}
    2c_1e^{2x}&3c_2e^{3x}\\
    4c_1e^{2x}&3c_2e^{3x}
    \end{bmatrix}$


    hopefully
    typos maybe
    Last edited by bigwave; Mar 29th 2019 at 05:17 PM.
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  4. #4
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    Re: 6.1.1 Show that .... is a solution of.....

    Yes, that is correct!
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  5. #5
    Super Member bigwave's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    These are too many tedious steps
    Ok for 6.1.2 I'm not sure just what Y is
    Since it's a different problem
    Do we take the integral of Y' if so how?
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  6. #6
    Forum Admin topsquark's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    Quote Originally Posted by bigwave View Post
    These are too many tedious steps
    Ok for 6.1.2 I'm not sure just what Y is
    Since it's a different problem
    Do we take the integral of Y' if so how?
    You don't need to do anything further with this. They simply gave you an example of a problem and just made it up. It can get tedious but the method does make things easier, especially if you are dealing with several simultanous differential equations.

    Since you asked, the integral of a function $\displaystyle W = \left ( \begin{matrix} Y \\ Z \end{matrix} \right )$ is simply
    $\displaystyle \int W ~ dx = \int \left ( \begin{matrix} Y \\ Z \end{matrix} \right ) ~ dx = \left ( \begin{matrix} \int Y ~ dx \\ \int Z ~ dx \end{matrix} \right )$

    You take the integral of the components just like to find the derivative of W you simply take the derivative of the components.

    -Dan
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  7. #7
    Super Member bigwave's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    Really appreciate the help
    I'll continue on this on Monday
    When I use the university PC
    I just have a cell phone on weekends
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  8. #8
    Super Member bigwave's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    so how do put in the "solved" prefix? I don't see that option in thread tools?
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  9. #9
    Forum Admin topsquark's Avatar
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    Re: 6.1.1 Show that .... is a solution of.....

    Quote Originally Posted by bigwave View Post
    so how do put in the "solved" prefix? I don't see that option in thread tools?
    That feature was removed at some point. And for some silly reason spammers can do that, so it's still in the system somewhere.

    -Dan
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