MATH 308 Lecture 23
« previous | Monday, March 18, 2013 | next »
Exam Review
Fundamental Sets of Solutions
Recall 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 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 a fundamental set of solutions 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 W(y_1, y_2) \ne 0} .
Reduction of Order
Given a differential equation and one 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} , we want to find a second solution in 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_2 = y_1 \, \lambda}
For example, given 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'' - x\,y' + y = 0} 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 = \mathrm{e}^{x}} , we find the second solution as follows:
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 \, \mathrm{e}^{x} \\ y_2' &= \lambda' \, \mathrm{e}^{x} + \lambda \, \mathrm{e}^{x} \\ y_2'' &= \lambda'' \, \mathrm{e}^{x} + 2 \lambda' \, \mathrm{e}^{x} + \lambda \, \mathrm{e}^{x} \\ 0 &= (x-1) \, \left( \lambda'' + 2 \lambda' + \lambda \right) \, \mathrm{e}^{x} - x\,(\lambda + \lambda')\,\mathrm{e}^{x} + \lambda \, \mathrm{e}^{x} \\ 0 &= (x-1) \, \lambda'' + (x-2)\, \lambda' + \cancel{0 \lambda} \\ \mu &= \mathrm{e}^{\int \frac{x-2}{x-1} \, \mathrm{d}x} = \mathrm{e}^{\int 1 - \frac{1}{x-1} \, \mathrm{d}x} = \frac{\mathrm{e}^{x}}{x-1} \\ \frac{\mathrm{d}}{\mathrm{d}x} \left( \lambda' \, \frac{\mathrm{e}^{x}}{x-1} \right) &= 0 \\ \lambda' \, \frac{\mathrm{e}^{x}}{x-1} &= C \\ \lambda' &= C \, (x-1) \, \mathrm{e}^{-x} \\ \lambda &= -C \, ((x+1) \, \mathrm{e}^{-x} - \mathrm{e}^{-x}) \end{align}}
Variation of Parameters
Recall that for non-homogeneous differential equations, a particular solution may be found by solving for the homogeneous solution, and then using the corresponding 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} in the formulae:
Laplace Transforms
Review starting with MATH 308 Lecture 17.