Derin Hames 2017a - Revision history

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Rimni: Rimni moved page Draft Samper 135489180 to Derin Hames 2017a

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Rimni moved page Draft Samper 135489180 to Derin Hames 2017a

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← Older revision		Revision as of 10:29, 20 September 2017
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−	== 2. A Simple Approach to the Particular Solution, <~~math display="inline"~~> ~~\mathbf{y_{~~p}} </~~math~~>==	+	== 2. A Simple Approach to the Particular Solution, y<sub>p</sub>==

	Particular solution of considered differential equations which is linear, constant coefficient, non-homogeneous and higher order differential equation is written as		Particular solution of considered differential equations which is linear, constant coefficient, non-homogeneous and higher order differential equation is written as

@@ Line 341: / Line 341: @@
 <math display="inline">y_{p}=\frac{1}{\left(-2ia\right)^{r}F_{1}(-ia)}\left(e^{-iax}\int  e^{iax}e^{-iax}\frac{x^{r-1}}{(r-1)!}dx\right)=\frac{x^{r}}{r!}\frac{e^{-iax}}{\left(-2ia\right)^{r}F_{1}(-ia)}</math>.
 ==2. A Simple Approach to the Particular Solution, <math>\mathbf{y_{p}}</math>==
 Particular solution of considered differential equations which is linear, constant coefficient, non-homogeneous and higher order differential equation is written as

@@ Line 524: / Line 524: @@
 Consider such a differential equation
-<math display="inline">(D^{2}+1)^{10}y=2\sin x-3\cos x</math> denkleminin çözümü.
+<math display="inline">(D^{2}+1)^{10}y=2\sin x-3\cos x</math>.
 Since <math display="inline">n=10</math>, the method recommended for the even value in equation (14) is used. Therefore,

@@ Line 341: / Line 341: @@
 <math display="inline">y_{p}=\frac{1}{\left(-2ia\right)^{r}F_{1}(-ia)}\left(e^{-iax}\int  e^{iax}e^{-iax}\frac{x^{r-1}}{(r-1)!}dx\right)=\frac{x^{r}}{r!}\frac{e^{-iax}}{\left(-2ia\right)^{r}F_{1}(-ia)}</math>.
-==2 A Simple Approach to the Particular Solution, <math>\mathbf{y_{p}}</math>==
+==2. A Simple Approach to the Particular Solution, <math>\mathbf{y_{p}}</math>==
 Particular solution of considered differential equations which is linear, constant coefficient, non-homogeneous and higher order differential equation is written as
 <span id="(eq-10)"></span>
@@ Line 441: / Line 442: @@
 |}
-==3 Examples==
+==3. Examples==
 === Example 1 ===
@@ Line 563: / Line 564: @@
 +120D_{x^{6}}y+45D_{x^{4} }y+10D_{x^{2}}y+y  ={2}\sin x-3\cos x.\end{array}</math>
-==4 Concluding remarks==
+==4. Concluding remarks==
 When using the classical analytical methods known in the literature for solving the high order linear differential equations, the resulting systems of integral and multivariable linear equations make the numerical calculations of the related physical problems difficult and complex algorithms. However, the method we propose above and proved its validity will remove these difficulties altogether. This simple analytical solution we have presented will help to create a rapid numerical algorithm for computers and thus simplify the numerical solution of high order physical problems.
-===Bibliography===
+===References===
 <div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
 <div id="cite-1"></div>

@@ Line 3: / Line 3: @@
 We suggest a practical method for obtaining the particular solution of non-homogeneous higher order linear differential equations with constant coefficients. The proposed method can be applied directly and simply to such problems. We revealed that is valid for the different type of problem by using sample solutions. This simple analytical solution that we have introduced will help to create a fast numerical algorithm for computers and thus simplify the numerical solutions of higher order physical problems.
-==1 Introduction==
+==1. Introduction==
 Differential equations are vital argument for technical and fundamental sciences. Ordinary differential equations with constant coefficients provide practical utility owing to their mathematical controllable  to  explain  natural  phenomena  in  many  areas  of  science  and  engineering <span id='citeF-1'></span><span id='citeF-2'></span>[[#cite-1|[1,2]]]. For example, they are widely used to explain the event of electromagnetic, sound and water wave.  Such  waves are  formulated  by  homogeneous  or  non-homogeneous  differential  equations  with higher  order  linear  constant  coefficients  depend  upon  source  of  the  phenomena <span id='citeF-3'></span><span id='citeF-4'></span><span id='citeF-5'></span><span id='citeF-6'></span><span id='citeF-7'></span><span id='citeF-8'></span><span id='citeF-9'></span><span id='citeF-10'></span>[[#cite-3|[3,4,5,6,7,8,9,10]]].

← Older revision	Revision as of 09:19, 30 August 2017
(No difference)

← Older revision		Revision as of 11:14, 29 August 2017
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	We suggest a practical method for obtaining the particular solution of non-homogeneous higher order linear differential equations with constant coefficients. The proposed method can be applied directly and simply to such problems. We revealed that is valid for the different type of problem by using sample solutions. This simple analytical solution that we have introduced will help to create a fast numerical algorithm for computers and thus simplify the numerical solutions of higher order physical problems.		We suggest a practical method for obtaining the particular solution of non-homogeneous higher order linear differential equations with constant coefficients. The proposed method can be applied directly and simply to such problems. We revealed that is valid for the different type of problem by using sample solutions. This simple analytical solution that we have introduced will help to create a fast numerical algorithm for computers and thus simplify the numerical solutions of higher order physical problems.
−

	==1 Introduction==		==1 Introduction==

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 |}
-==Proof.==
+'''Proof.'''
 Let <math display="inline">F(D)y=e^{-iax}</math> Therefore,