Difference between revisions of "Reduction of order"
(Created page with ''''Reduction of order''' is a process used to find the solution of a differential equation <math>y'' + p(t)y' + q(t)y = 0</math>, when Euler substitution methods find only on…') |
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1. <math>y_1 = e^{\lambda t}</math>. However, we must find <math>y_2</math>. <math>y_2</math> cannot be a multiple of <math>y_1</math>, so we assume that <math>y_2 = v(t) y_1</math>. | 1. <math>y_1 = e^{\lambda t}</math>. However, we must find <math>y_2</math>. <math>y_2</math> cannot be a multiple of <math>y_1</math>, so we assume that <math>y_2 = v(t) y_1</math>. | ||
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2. We apply the product rule to find: | 2. We apply the product rule to find: | ||
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<math>y_2'' = v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1''</math> | <math>y_2'' = v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1''</math> | ||
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3. We substitute these expressions into the initial differential equation: | 3. We substitute these expressions into the initial differential equation: | ||
<math>y_2'' + p(t)y_2' + q(t)y = (v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1'') + p(t)(v(t)' y_1 + v(t) y_1') + q(t)(v(t) y_1)</math> | <math>y_2'' + p(t)y_2' + q(t)y = (v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1'') + p(t)(v(t)' y_1 + v(t) y_1') + q(t)(v(t) y_1)</math> | ||
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4. We collect the <math>v(t)''</math>, <math>v(t)'</math>, and <math>v(t)</math> terms: | 4. We collect the <math>v(t)''</math>, <math>v(t)'</math>, and <math>v(t)</math> terms: | ||
<math>v(t)''(y_1) + v(t)'(2y_1' + p(t) y_1) + v(t)(y_1'' + p(t) y_1' + q(t) v(t) y_1)</math> | <math>v(t)''(y_1) + v(t)'(2y_1' + p(t) y_1) + v(t)(y_1'' + p(t) y_1' + q(t) v(t) y_1)</math> | ||
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5. The terms <math>(2y_1' + p(t) y_1)</math> and <math>(y_1'' + p(t) y_1' + q(t) v(t) y_1)</math> equal zero in most instances, leading to the conclusion: | 5. The terms <math>(2y_1' + p(t) y_1)</math> and <math>(y_1'' + p(t) y_1' + q(t) v(t) y_1)</math> equal zero in most instances, leading to the conclusion: | ||
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<math>v(t) = c_1 t + c_2</math> | <math>v(t) = c_1 t + c_2</math> | ||
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6. This is enough to say that <math>y_2 = v(t) y_1 = ty_1 = t e^{\lambda t}</math> | 6. This is enough to say that <math>y_2 = v(t) y_1 = ty_1 = t e^{\lambda t}</math> | ||
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7. The solution is then <math>y = c_1 y_1 + c_2 y_2 = e^{\lambda t} (c_1 + c_2 t)</math> | 7. The solution is then <math>y = c_1 y_1 + c_2 y_2 = e^{\lambda t} (c_1 + c_2 t)</math> | ||
[[Category: Differential Equations]] | [[Category: Differential Equations]] | ||
Revision as of 17:10, July 19, 2010
Reduction of order is a process used to find the solution of a 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'' + p(t)y' + q(t)y = 0} , when Euler substitution methods find only one value for . (That is, and .)
The process is carried out in the following manner:
1. . However, we must 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_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_2} cannot be a multiple 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_1} , so we assume 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_2 = v(t) y_1} .
2. We apply the product rule to 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_2 = v(t) 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 y_2' = v(t)' y_1 + v(t) 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 y_2'' = v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1''}
3. We substitute these expressions into the initial 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_2'' + p(t)y_2' + q(t)y = (v(t)'' y_1 + 2 v(t)' y_1' + v(t) y_1'') + p(t)(v(t)' y_1 + v(t) y_1') + q(t)(v(t) y_1)}
4. We collect 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 v(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 v(t)'}
, 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 v(t)}
terms:
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 v(t)''(y_1) + v(t)'(2y_1' + p(t) y_1) + v(t)(y_1'' + p(t) y_1' + q(t) v(t) y_1)}
5. The terms 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 (2y_1' + p(t) 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_1'' + p(t) y_1' + q(t) v(t) y_1)}
equal zero in most instances, leading to the conclusion:
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 v(t)''(y_1) = 0}
Integrating twice, we yield:
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 v(t) = c_1 t + c_2}
6. This is enough to say 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_2 = v(t) y_1 = ty_1 = t e^{\lambda t}}
7. The solution is 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 y = c_1 y_1 + c_2 y_2 = e^{\lambda t} (c_1 + c_2 t)}