MATH 308 Lecture 26

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Lecture Notes

Systems of Differential Equations

${\begin{cases}x_{1}'(t)=-2x_{1}+x_{2}\\x_{2}'(t)=x_{1}-2x_{2}\end{cases}}$

1. solve first equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2}
2. substitute into the second equation, thereby obtaining a second order equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1}
3. solve equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1}
4. determine Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x_2 &= x_1' + 2x_1 \\ \frac{\mathrm{d}}{\mathrm{d}x} \left( x_1' + 2x_1 \right) &= x_1 - 2 \left( x_1' + 2x_1 \right) \\ 0 &= x_1'' + 4x_1' + 3x_1 \\ x_1 &= c_1 \, \mathrm{e}^{-t} + c_2 \, \mathrm{e}^{-3t} \\ x_2 &= \left( -c_1 \, \mathrm{e}^{-t} - 3c_2 \, \mathrm{e}^{-3t} \right) + 2 \left( c_1 \, \mathrm{e}^{-t} + c_2 \, \mathrm{e}^{-3t} \right) \\ &= c_1 \, \mathrm{e}^{-t} - c_2 \, \mathrm{e}^{-3t} \end{align}}

Satisfying Initial Conditions

Find a particular solution of the system above that also satisfies the initial conditions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1(0) = 2} , $x_{2}(0)=3$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} x_1(0) = c_1 + c_2 = 2 \\ x_2(0) = c_1 - c_2 = 3 \end{cases}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 = \tfrac{5}{2}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 = -\tfrac{1}{2}} , so the particular solutions are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} x_1 = \frac{5}{2} \, \mathrm{e}^{-t} - \frac{1}{2} \, \mathrm{e}^{-3t} \\ x_2 = \frac{5}{2} \, \mathrm{e}^{-t} + \frac{1}{2} \, \mathrm{e}^{-3t} \end{cases}}

Matrix Notation

(See MATH 323 Lecture 26#Matrix Exponential→)

The previous system of equations can be represented as the matrix equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix}x_1 \\ x_2 \end{bmatrix}' = \begin{bmatrix}-2 & 1 \\ 1 & -2 \end{bmatrix} \, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}}

In general, we now have the equations represented as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X' = A \, X} , and the solution will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \left( \mathrm{e}^{A\,t} \right) \, X_0} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X' = A \, X \quad \iff \quad X = \left( \mathrm{e}^{A\,t} \right) \, X_0}

Exercise 2

Express the system of differential equations in matrix notation:

${\begin{cases}x_{1}'(t)=2t\,x_{1}(t)+5x_{3}(t)\\x_{2}'(t)=\left(\sin {t}\right)\,x_{1}(t)+t^{2}\,x_{2}(t)\\x_{3}'(t)=-x_{1}(t)+x_{2}(t)+3x_{3}(t)\end{cases}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}' = \begin{bmatrix} 2t & 0 & 5 \\ \sin{t} & t^2 & 0 \\ -1 & 1 & 3 \end{bmatrix} \, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}}

Exercise 3

Transform the differential equation into a system of first order equations. Express the system in matrix notation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + 3y' - 5y = \cos{3x}}

1. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 = y} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1' = y'}
2. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 = y'} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1' = x_2} (from above) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2' = y'' = 5y - 3y' + \cos{3x} = 5x_1 - 3x_2 + \cos{3x}}

Therefore, our system is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} x_1' = x_2 \\ x_2' = 5x_1 - 3x_2 + \cos{3x} \end{cases}}

This can be expressed as the matrix equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}' = \begin{bmatrix}0&1\\5&-3\end{bmatrix} \, \begin{bmatrix}x_1\\x_2\end{bmatrix} + \begin{bmatrix}0\\\cos{3x}\end{bmatrix}}

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Transform $y^{(4)}+3y'=5$ into a system of first order equations.

1. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 = y} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1' = y'}
2. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 = y'} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1' = x_2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2' = y''}
3. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_3 = y''} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2' = x_3} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_3' = y^{(3)}}
4. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_4 = y^{(3)}} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_3' = x_4} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_4' = y^{(4)} = -3y' + 5 = -3x_2 + 5}

Therefore, our system is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} x_1' = x_2 \\ x_2' = x_3 \\ x_3' = x_4 \\ x_4' = -3x_2 + 5 \end{cases}}

and can be expressed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}' = \begin{bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&-3&0&0\end{bmatrix} \, \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix} + \begin{bmatrix}0\\0\\0\\5\end{bmatrix}}