MATH 308 Lecture 17
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An Initial Value Problem
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'' - 2y' = 4x \quad y(0) = 0 \quad y'(0) = 0}
Solution to homogeneous problem 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''-2y'=0} 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 r \in \{0,2\} \rightarrow y_h = c_1 \, \cancel{\mathrm{e}^{0x}} + c_2 \, \mathrm{e}^{2x}}
Try method of undetermined coefficients (could also use method of variation of parameters)
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\,x + B} , but 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 \propto B} is already a solution,so 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 \, (A \, x + B)}
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_p &= A \, x^2 + B \,x \\ y_p' &= 2A \, x + B \\ y_p'' &= 2A \\ 4x &= y'' - 2y' \\ &= 2A - 2(2A \, x + B) \\ &= (-4A) \, x + (2A - 2B) \\ (A,B) &= (-1,-1) \end{align}}
So 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^2-x} is a particular solution, and our 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 y = y_p + y_h = -x^2-x+c_1+c_2\,\mathrm{e}^{2x}}
Plug in initial values and 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 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} :
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 = -x^2-x-\frac{1}{2} + \frac{1}{2} \, \mathrm{e}^{2x}}
Laplace Transforms
Piecewise functions that need not be continuous but must be piecewise continuous.
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 f} be a function on 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 \left[0, \infty \right)} . The Laplace transform 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 f} is the function 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 \mathcal{L}\left\{f\right\}} defined by the integral
The domain 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 \mathcal{L}\left\{f\right\}} is all the values 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 s} for which the integral exists.
The limit exists if 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 t > M} : 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} \left| f(t) \right| &\le k \mathrm{e}^{a \, t} \left| f(t) \, \mathrm{e}^{-s \, t} \right| &\le k \, \mathrm{e}^{(a-s)t} \end{align}}
So the integral 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 \int_0^\infty \mathrm{e}^{(a-s)t} \, \mathrm{d}t} is convergent 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 s > a}
Our goal is to be able to take the laplace transform of each side 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 y'' + y = g(t)} :
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 \mathcal{L}\left\{y''\right\}(s) + \mathcal{L}\left\{y\right\}(s) = \mathcal{L}\left\{g(t)\right\}(s)}
Piecewise Continuity
A function is piecewise continuous on an interval 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 a \le t \le b} if the interval 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 \left[ a, b \right]} can be partitioned by a finite number of points 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 a = t_0 < t_1 < \dots < t_n = b} so that
- 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 f} is continuous on each open interval 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 \left( t_i, t_{i+1} \right)} .
- 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 f} approached a finite limit as the endpoints of each interval are approached from within the subinterval.
Exercises
Find the Laplace transform 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 f(t) = 1} :
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 \mathcal{L} \left\{ 1 \right\}(s) = \int_0^\infty \mathrm{e}^{-s\,t} \, \mathrm{d}t}
- Does not exist 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 s \le 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 \begin{align} \mathcal{L} \left\{ 1 \right\}(s) &= \lim_{N \to \infty} \int_0^N \mathrm{e}^{-s\,t} \, \mathrm{d}t \\ &= \lim_{N \to \infty} \frac{\mathrm{e}^{-s \, N}}{-s} - \frac{1}{-s} \\ &= \lim_{N \to \infty} -\frac{\mathrm{e}^{-s \, N}}{s} + \frac{1}{s} \\ &= \frac{1}{s} \quad s > 0 \end{align}}
Find the Laplace transform 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 f(t) = t} :
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 \mathcal{L} \left\{ t \right\}(s) = \int_0^\infty t \, \mathrm{e}^{-s\,t} \, \mathrm{d}t}
- Does not exist 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 s \le 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 \begin{align} \mathcal{L} \left\{ 1 \right\}(s) &= \lim_{N \to \infty} \int_0^N t \, \mathrm{e}^{-s\,t} \, \mathrm{d}t \\ &= \lim_{N \to \infty} \left. \frac{-t \, \mathrm{e}^{-s\,t}}{s} \right|_0^N + \int_0^N \frac{\mathrm{e}^{-s\,t}}{s} \, \mathrm{d}t \\ &= \lim_{N \to \infty} \frac{-N \, \mathrm{e}^{-s\,N}}{s} - \frac{\mathrm{e}^{-s \, N}}{s^2} + \frac{1}{s^2} \\ &= \frac{1}{s^2} \quad s > 0 \end{align}}
Find the Laplace transform 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 f(t) = \mathrm{e}^{3t}}
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 \mathcal{L} \left\{ \mathrm{e}^{3t} \right\}(s) = \lim_{N \to \infty} \int_0^N \mathrm{e}^{(3-s)t} \, \mathrm{d}t}
- 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 3-s > 0} , so integral exists 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 s > 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{align} \lim_{N \to \infty} \int_0^N \mathrm{e}^{(3-s)t} \, \mathrm{d}t &= \left. \frac{\mathrm{e}^{(3-s)t}}{3-s} \right|_0^N \\ &= \frac{1}{s-3} \quad s > 3 \end{align}}
Find the Laplace transform 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 f(t) = \mathrm{e}^{5t} \, \cos{t}}
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 \mathcal{L} \left\{ \mathrm{e}^{5t} \, \cos{t} \right\} = \lim_{N \to \infty} \int_0^N \mathrm{e}^{(5-s)t} \, \mathrm{d}t}
- Exists for
Integrate by parts:
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} \lim_{N \to \infty} \int_0^N \mathrm{e}^{(5-s)t} \, \mathrm{d}t &= \lim_{N\to\infty} \left. \mathrm{e}^{(5-s)t} \, \sin{t} \right|_0^N - \int_0^N (5-s) \, \mathrm{e}^{(5-s)t} \, \mathrm{d}t \\ &= \lim_{n\to\infty} 0 - \left. \mathrm{e}^{(5-s)t} \, \cos{t} \right|_0^N - \int_0^N (5-s)^2 \, \mathrm{e}^{(5-s)t} \, \cos{t} \, \mathrm{d}t \\ &= -(5-s) - (5-s)^2 \, \mathcal{L} \left\{ \mathrm{e}^{5t} \, \cos{t} \right\} \\ &= \frac{s-5}{1+(s-5)^2} \end{align}}