MATH 308 Lecture 15

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Lecture Notes

Remark for section 3.5

How would we go about solving $y''-3y'+2y=20\sin {2x}+2x+3+\mathrm {e} ^{x}$?

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_h = c_1 \, \mathrm{e}^{2x} + c_2 \, \mathrm{e}^{x}}

Particular Solution 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_p = A \, \sin{2x} + B \, \cos{2x} + a\,x + b + C\,x\,\mathrm{e}^{x}}

Section 3.6: Variation of Parameters

Find solution to $y''-y'-2y=\mathrm {e} ^{3x}$

Homogeneous solution 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 y_h = c_1 \, \mathrm{e}^{-x} + c_2 \, \mathrm{e}^{2x}}

What about non-constant coefficients; i.e. what if the constants were actually functions of 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} :

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 = c_1(x) \, y_1 + c_2(x) \, y_2}

There are infinitely many possibilities 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 c_1} 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} , so we impose a restriction to limit the number of possibilities:

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' \, y_1 + c_2' \, y_2 = 0}

Using the above examples, we 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 y_1 = \mathrm{e}^{-x}} and $y_{2}=\mathrm {e} ^{2x}$:

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} y &= c_1 \, \mathrm{e}^{-x} + c_2 \, \mathrm{e}^{2x} \\ y' &= c_1' \, \mathrm{e}^{-x} - c_1 \, \mathrm{e}^{-x} + c_2' \, \mathrm{e}^{2x} + 2c_2 \, \mathrm{e}^{2x} \\ &= \left( c_1' \, \mathrm{e}^{-x} + c_2' \, \mathrm{e}^{2x} \right) -c_1 \, \mathrm{e}^{-x} + 2c_2 \, \mathrm{e}^{2x} \\ &= -c_1 \, \mathrm{e}^{-x} + 2c_2 \, \mathrm{e}^{2x} \\ y'' &= c_1 \, \mathrm{e}^{-x} - c_1' \, \mathrm{e}^{-x} + 2c_2' \, \mathrm{e}^{2x} + 4c_2 \, \mathrm{e}^{2x} \end{align}}

Simplification: We were able to simplify 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'} with the restriction 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' \, \mathrm{e}^{-x} + c_2' \, \mathrm{e}^{2x} = 0} .

Now we can plug our functions into the differential 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{align} \mathrm{e}^{3x} &= y'' - y' -2y \\ &= (c_1 \, \mathrm{e}^{-x}-c_1' \, \mathrm{e}^{-x} + 2c_2' \, \mathrm{e}^{2x} + 4c_2 \, \mathrm{e}^{2x}) - (-c_1 \, \mathrm{e}^{-x} + 2c_2 \, \mathrm{e}^{2x}) - 2(c_1\,\mathrm{e}^{-x} + c_2\,\mathrm{e}^{2x}) \\ &= \cancel{c_1 \, \mathrm{e}^{-x}} -c_1' \, \mathrm{e}^{-x} + 2c_2' \, \mathrm{e}^{2x} + \cancel{4c_2 \, \mathrm{e}^{2x}} + \cancel{c_1 \, \mathrm{e}^{-x}} \cancel{-2c_2 \, \mathrm{e}^{2x}} \cancel{- 2c_1 \, \mathrm{e}^{-x}} \cancel{-2c_2\,\mathrm{e}^{2x}} \\ &= -c_1' \, \mathrm{e}^{-x} + 2c_2' \, \mathrm{e}^{2x} \end{align}}

Use restriction as second equation in system:

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} -c_1' \, \mathrm{e}^{-x} + 2c_2' \, \mathrm{e}^{2x} = \mathrm{e}^{3x} \\ c_1' \, \mathrm{e}^{-x} + c_2' \, \mathrm{e}^{2x} = 0 \end{cases}}

We solve 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 c_1'} and $c_{2}'$:

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} c_1' &= -\frac{1}{3} \, \mathrm{e}^{4x} & c_2' &= \frac{1}{3} \, \mathrm{e}^{x}\\ c_1 &= -\frac{1}{12} \, \mathrm{e}^{4x} + A & c_2 &= \frac{1}{3} \, \mathrm{e}^{x} + B \end{align}}

So general solution to original differential 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 y'' - y' - 2y = \mathrm{e}^{3x}} 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{align} y &= \left( -\frac{1}{12} \, \mathrm{e}^{4x} + A \right) \mathrm{e}^{-x} + \left( \frac{1}{3} \, \mathrm{e}^{x} + B \right) \mathrm{e}^{2x} \\ &= \left( -\frac{1}{12} \, \mathrm{e}^{3x} + \frac{1}{3} \, \mathrm{e}^{3x} \right) + A \, \mathrm{e}^{-x} + B \, \mathrm{e}^{2x} \\ &= \frac{1}{4} \, \mathrm{e}^{3x} + A \, \mathrm{e}^{-x} + B \, \mathrm{e}^{2x} \end{align}}

Method for Solving Second Order ODEs

To determine a particular solution to 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'' + p(x) \, y' + q(x) \, y = g(x)}

Find 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_1} and $y_{2}$ for corresponding homogeneous 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 y'' + p(x) \, y' + q(x) \, y = 0}

Solve the following system 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 c_1(x)} 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(x)}

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} c_1' \, y_1 + c_2' \, y_2 = 0 \\ c_1' \, y_1' + c_2' \, y_2' = g(x) \end{cases}}

or

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} c_1(x) &= - \int_{x_0}^x \frac{y_2(s) \, g(s)}{W(y_1,y_2)(s)} \, \mathrm{d}s & c_2(x) &= \int_{x_0}^x \frac{y_1(s) \, g(s)}{W(y_1,y_2)(s)} \, \mathrm{d}s \end{align}}

A particular 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 y_p(x) = c_1(x) \, y_1(x) + c_2(x) \, y_2(x)}

Exercise 7

Find general solution to $y''+y=0$

Solution to homogeneous equation 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 y = c_1 \, \cos{x} + c_2 \, \sin{x}} , where 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_1 = \cos{x}} 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 y_2 = \sin{x}} .

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 W(y_1,y_2) = \cos^2(x) + \sin^2(x) = 1}

Plug in to equations 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 c_1} and $c_{2}$

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} c_1(x) &= -\int \sin{x} \, \frac{1}{\sin{x}} \, \mathrm{d}x \\ &= -x + A \\ c_2(x) &= \int \cos{x} \, \frac{1}{\sin{x}} \, \mathrm{d}x \\ &= \ln{(\sin{x})} + B \end{align}}

General solution 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{align} y &= \left( -x + A \right) \, \cos{x} + \left( \ln{(\sin{x})} + B \right) \sin{x} \\ &= -x \, \cos{x} + A\,\cos{x} + \ln{(\sin{x})} \, \sin{x} + B\,\sin{x} \end{align}}