MATH 308 Lecture 13
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Wronskian
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) = y_1 \, y_2' - y_2 \, y_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 W(y_1, y_2) \ne 0} iff 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 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} form a fundamental set of solutions. Any solution is of form 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 \, y_1 + c_2 \, y_2}
Abel's Theorem
Exercise 14
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(t) \, y = 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 y_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 y_2} are two solutions, and they reach the maximum at the same point 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_0} . Are they a fundamental set of solutions?
Evaluate 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)(t_0) = y_1(t_0) \, y_2'(t_0) - y_2(t_0) \, y_1'(t_0) = y_1(t_0) \cdot 0 - y_2(t_0) \cdot 0 = 0} .
Therefore, since 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) = 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 y_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 y_2} cannot be a fundamental set of solutions.
Chapter 3.4: Reduction of Order method
Given linear, homogeneous differential equation and 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_1} :
Find solutions of the form 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) = \lambda(x) \, y_1(x)}
Differentiate 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} as many times as needed:
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' &= \lambda' \, y_1 + \lambda \, y_1' \\ y'' &= \lambda'' \, y_1 + 2 \lambda' \, y_1 + \lambda \, y_1'' \\ y^{(n)} &= \ldots \end{align}}
Plug the derivatives into the original differential equation. The final term should always 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 0 \lambda} .
Solve the reduced order differential 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 \lambda'} and plug back into original 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} .
Exercise 15
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'' - 5y' + 6y = 0}
We are told 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 e^x} is a solution, but we want to find more solutions of the form 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 = \lambda(x) \, \mathrm{e}^x} , where the 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 \mathrm{e}^x} is the 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 \begin{align} y'(x) &= \lambda'(x) \, \mathrm{e}^{x} + \lambda(x) \, \mathrm{e}^{x} \\ y''(x) &= \lambda''(x) \, \mathrm{e}^{x} + 2\lambda'(x) \, \mathrm{e}^{x} + \lambda(x) \mathrm{e}^{x} \\ y'''(x) &= \lambda'''(x) \, \mathrm{e}^{x} + 3 \lambda''(x) \, \mathrm{e}^x, + 3 \lambda'(x) \, \mathrm{e}^x + \lambda(x) \, \mathrm{e}^x \end{align}}
Therefore, our differential equation becomes
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 \lambda'''(x) \, \mathrm{e}^x + 3\lambda''(x) \, \mathrm{e}^x + 3 \lambda'(x) \, \mathrm{e}^{x} + \lambda(x) \, \mathrm{e}^{x} + 2 \left( \lambda''(x) \, \mathrm{e}^{x} + 2\lambda'(x) \, \mathrm{e}^{x} + \lambda(x) \mathrm{e}^{x} \right) - 5 \left( \lambda'(x) \, \mathrm{e}^{x} + \lambda(x) \, \mathrm{e}^{x} \right) + 5 \lambda(x) \, \mathrm{e}^x = 0}
This can be reduced 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 \begin{align} \lambda'''(x) \, \mathrm{e}^{x} + \lambda'' \, \mathrm{e}^{x} - 6 \lambda'(x) \, \mathrm{e}^{x} + 0 \lambda &= 0 \\ \lambda'''(x) + \lambda''(x) - 6 \lambda'(x) &= 0 \end{align}}
The 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 0 \lambda} is always the case. Notice 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 \lambda'(x)} is a solution to a second order differential equation. 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(x) = \lambda'(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'' + y' - 6y = 0}
Characteristic 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 r^2 + r - 6 = 0} , roots 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 r \in \{-3,2 \}} .
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} \lambda' &= c_1 \, \mathrm{e}^{-3t} + c_2 \, \mathrm{e}^{2t} \\ \lambda &= -\frac{c_1}{3} \, \mathrm{e}^{-3t} + \frac{c_2}{2} \, \mathrm{e}^{2x} + C \end{align}}
Therefore, our 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 = \lambda(x) \, \mathrm{e}^{x} = \left( -\frac{c_1}{3} \right) \, \mathrm{e}^{-2x} + \left( \frac{c_2}{2} \right) \, \mathrm{e}^{3x} + C \, \mathrm{e}^x}
Remark
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'' - 4y' + 4y = 0}
Characteristic equation has root 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 = 2} with multiplicity 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_1 = \mathrm{e}^{2x}}
Find second 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_2 = \lambda(x) \, \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_2' &= \lambda'(x) \, \mathrm{e}^{2x} + 2 \lambda'(x) \, \mathrm{e}^{2x} \\ y_2'' &= \lambda''(x) \, \mathrm{e}^{2x} + 4 \lambda'(x) \, \mathrm{e}^{2x} + 4 \lambda(x) \, \mathrm{e}^{2x} \end{align}}
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} 0 &= y_2'' - 4y_2' + y_2 \\ &= \left( \lambda''(x) \, \mathrm{e}^{2x} + 4 \lambda'(x) \, \mathrm{e}^{2x} + 4 \lambda(x) \, \mathrm{e}^{2x} \right) + 4 \left( \lambda'(x) \, \mathrm{e}^{2x} + 2 \lambda'(x) \, \mathrm{e}^{2x} \right) + 4 \lambda(x) \, \mathrm{e}^{2x} \end{align}}
Therefore 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 \lambda''(x) \, \mathrm{e}^{2x} = 0} , giving 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 \lambda(x) = a\,x + b}