Solutions of Differential Equations - Revision history

Khanh at 05:38, 6 November 2021

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Khanh at 05:32, 6 November 2021

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Lila at 01:57, 18 September 2021

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Lila at 01:54, 18 September 2021

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Lila: Created page with "A solution of a differential equation is an expression of the dependent variable that satisfies the relation established in the differential equation. For example, the solutio..."

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Created page with "A solution of a differential equation is an expression of the dependent variable that satisfies the relation established in the differential equation. For example, the solutio..."

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A solution of a differential equation is an expression of the dependent variable that satisfies the relation established in the differential equation. For example, the solution of <math> y' + y - 2 = 4x </math> will be some equation y = f(x) such that y and its first derivative, y', satisfy the relation <math> y' + y - 2 = 4x </math>. The general solution of a differential equation will have one or more arbitrary constants, depending on the order of the original differential equation (the solution of a first order diff. eq. will have one arbitrary constant, a second order one will have two, etc.).

Examples:
* <math> y' = 2x </math>. Through simple integration, we can calculate the general solution of this equation to be <math>y = x^2 + C</math>, where C is an arbitrary constant.
* <math> y' - y = 0 </math>. The G.S. is <math> y = Ce^{x} </math>. <math> y' = Ce^{x} </math>, so <math> y' - y = 0 \to Ce^{x} - Ce^{x} = 0</math>, so this solution satisfies the relationship for all arbitrary constants C.
* <math> y' + y - 2 = 0 </math>. The G.S. is <math> y = Ce^{-x} + 2 </math>. <math> y' = -Ce^{-x} </math>, so <math> 0 = y' + y - 2 </math> becomes <math> 0 = -Ce^{-x} + Ce^{-x} + 2 - 2 = 0 </math>.

==Resources==

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 The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants (see initial value problems for more).
 ==Resources==
 * [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations], University of Glascow
 * [https://www.api.simply.science/index.php/math/calculus/differential-equations/types-of-differential-equations/9869-general-and-particular-solutions-of-a-differential-equation General and Particular Solutions], Simply Math

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 * <math> y'' + y' - 2y = 0 </math>. The G.S. is <math> y = Ce^{x} + De^{-2x} </math>. <math> y' = Ce^{x} - 2De^{-2x} </math> and <math> y'' = Ce^{x} + 4De^{-2x} </math>, so <math> 0 = y'' + y' - 2y </math> becomes <math> 0 = Ce^{x} + 4De^{-2x} + Ce^{x} - 2De^{-2x} - 2(Ce^{x} + De^{-2x}) = Ce^{x} + Ce^{x} - 2Ce^{x} + 4De^{-2x} - 2De^{-2x} - 2De^{-2x}) = 0 </math>.
-The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants (see [[Initial Value Problem (IVP)| initial value problems]] for more).
+The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants (see initial value problems for more).
 ==Resources==
 * [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations], University of Glascow
 * [https://www.api.simply.science/index.php/math/calculus/differential-equations/types-of-differential-equations/9869-general-and-particular-solutions-of-a-differential-equation General and Particular Solutions], Simply Math

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 * <math> y'' + y' - 2y = 0 </math>. The G.S. is <math> y = Ce^{x} + De^{-2x} </math>. <math> y' = Ce^{x} - 2De^{-2x} </math> and <math> y'' = Ce^{x} + 4De^{-2x} </math>, so <math> 0 = y'' + y' - 2y </math> becomes <math> 0 = Ce^{x} + 4De^{-2x} + Ce^{x} - 2De^{-2x} - 2(Ce^{x} + De^{-2x}) = Ce^{x} + Ce^{x} - 2Ce^{x} + 4De^{-2x} - 2De^{-2x} - 2De^{-2x}) = 0 </math>.
-The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants.
+The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants (see [[Initial Value Problem (IVP)| initial value problems]] for more).
 ==Resources==
 * [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations], University of Glascow
 * [https://www.api.simply.science/index.php/math/calculus/differential-equations/types-of-differential-equations/9869-general-and-particular-solutions-of-a-differential-equation General and Particular Solutions], Simply Math

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 Examples:
 * <math> y' = 2x </math>, <math> y(2) = 0 </math>. With this point and the general solution <math>y = x^2 + C</math>, we can calculate the constant C to be -4. Thus the particular solution is <math>y = x^2 - 4</math>.
-* <math> y' - y = 0 </math>, <math> y(0) = 3 </math>. <math> 3 = Ce^{0} = C</math>, so the particular solution is <math> y = 3e^{x} </math>.
+* <math> y' - y = 0 </math>, <math> y(0) = 3 </math>. <math> 3 = Ce^{0} \implies C = 3</math>, so the particular solution is <math> y = 3e^{x} </math>.
 * <math> y'' + y' - 2y = 0 </math>, <math> y(0) = 2 </math>, <math> y'(0) = -1 </math>. So, <math> 2 = Ce^{0} + De^{0} = C + D </math> and <math> -1 = Ce^{0} - 2De^{0} = C - 2D</math>. Thus C = 1 and D = 1, and the particular solution is <math> y = e^{x} + e^{-2x} </math>.

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 * <math> y' = 2x </math>, <math> y(2) = 0 </math>. With this point and the general solution <math>y = x^2 + C</math>, we can calculate the constant C to be -4. Thus the particular solution is <math>y = x^2 - 4</math>.
 * <math> y' - y = 0 </math>, <math> y(0) = 3 </math>. <math> 3 = Ce^{0} = C</math>, so the particular solution is <math> y = 3e^{x} </math>.
-* <math> y'' + y' - 2y = 0 </math>. The G.S. is <math> y = Ce^{x} + De^{-2x} </math>. <math> y' = Ce^{x} - 2De^{-2x} </math> and <math> y'' = Ce^{x} + 4De^{-2x} </math>, so <math> 0 = y'' + y' - 2y </math> becomes <math> 0 = Ce^{x} + 4De^{-2x} + Ce^{x} - 2De^{-2x} - 2(Ce^{x} + De^{-2x}) = Ce^{x} + Ce^{x} - 2Ce^{x} + 4De^{-2x} - 2De^{-2x} - 2De^{-2x}) = 0 </math>.
+* <math> y'' + y' - 2y = 0 </math>, <math> y(0) = 2 </math>, <math> y'(0) = -1 </math>. So, <math> 2 = Ce^{0} + De^{0} = C + D </math> and <math> -1 = Ce^{0} - 2De^{0} = C - 2D</math>. Thus C = 1 and D = 1, and the particular solution is <math> y = e^{x} + e^{-2x} </math>.
 ==Resources==
 * [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations], University of Glascow
 * [https://www.api.simply.science/index.php/math/calculus/differential-equations/types-of-differential-equations/9869-general-and-particular-solutions-of-a-differential-equation General and Particular Solutions], Simply Math

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 * <math> y' = 2x </math>. Through simple integration, we can calculate the general solution of this equation to be <math>y = x^2 + C</math>, where C is an arbitrary constant.
 * <math> y' - y = 0 </math>. The G.S. is <math> y = Ce^{x} </math>. <math> y' = Ce^{x} </math>, so <math> y' - y = 0 \to Ce^{x} - Ce^{x} = 0</math>, so this solution satisfies the relationship for all arbitrary constants C.
-* <math> y' + y - 2 = 0 </math>. The G.S. is <math> y = Ce^{-x} + 2 </math>. <math> y' = -Ce^{-x} </math>, so <math> 0 = y' + y - 2 </math> becomes <math> 0 = -Ce^{-x} + Ce^{-x} + 2 - 2 = 0 </math>.
+* <math> y'' + y' - 2y = 0 </math>. The G.S. is <math> y = Ce^{x} + De^{-2x} </math>. <math> y' = Ce^{x} - 2De^{-2x} </math> and <math> y'' = Ce^{x} + 4De^{-2x} </math>, so <math> 0 = y'' + y' - 2y </math> becomes <math> 0 = Ce^{x} + 4De^{-2x} + Ce^{x} - 2De^{-2x} - 2(Ce^{x} + De^{-2x}) = Ce^{x} + Ce^{x} - 2Ce^{x} + 4De^{-2x} - 2De^{-2x} - 2De^{-2x}) = 0 </math>.

← Older revision		Revision as of 01:33, 18 September 2021
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	==Resources==		==Resources==
−	* [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations],	+	* [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations], University of Glascow
		+	* [https://www.api.simply.science/index.php/math/calculus/differential-equations/types-of-differential-equations/9869-general-and-particular-solutions-of-a-differential-equation General and Particular Solutions], Simply Math

← Older revision		Revision as of 01:30, 18 September 2021
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	==Resources==		==Resources==
		+	* [http://www.maths.gla.ac.uk/~cc/2x/2005_2xnotes/2x_chap5.pdf Differential Equations],