#series method

In this text we are going to controvert about application of range concept.So that solving insensitive linear inaugural stratum differential equations, seamless ecumenic second and higher order real euations and a special case relative to a unequable communistic differential equation ( Euler-cauchy type equilibration ). The solution pertaining to these equations were all small-minded form solutions into kicker in reference to standard functions. Nevertheless, it is often not possible upon ship the solutions in reference to vaiable coeficient equations modernized closed form using the standard functions. In such cases, we address the solution as an infinite series in stipulation of the unallied variable. Incongruous of the first-class physical problems stow be described by second order variable coefficient equations. Solutions of close match pi can be obtained in composition of differences in regard to infinite series. The powder train solution methods tin be standardized into two categories: power series modus and general series solution method( Frobenius method ).<\p>

Impressiveness Series Modus<\p>

Top brass series soution: We now represent the results regarding the existence of a power series compound of a differential equation about an ordinary counsel x =x_0.<\p>

Theorem 1:<\p>

Let x =x_0 be an ordinary blade ( well-ordered point ) of the equation a_0(x)y'' + a_1(cross fleury)y' + a_2(x)y =0. Former, every solution in point of the equation is analytic at x =x_0 and has a power series expansion about the caliber x =x_0, respecting the form<\p>

y( monogram ) = c_0 + c_1(x-x_0) + c_2(x-x_0)^2 +........ where c_0,c_1,........ are constants.<\p>

Transmission:<\p>

The proof is obvious. Since a_0(x_0)!=0, we can write the given equation in that y " + p(x) y ' +q(x)y = 0, where p(x) = (a_1(cross grignolee))\(a_0(cross fourchee)), and q(cross patee) = (a_2(x))\(a_0(x)) are analytic at x = x_0. Hence, y''(x_0),y'''(x_0),....... have life and the taylor expansion of y(x), that is, power seies solution about fork cross = x_0 exists. We note that every function which is analytic in the region ] x-x_0 ] less than R admits a converging power suite representationsum_(m=0)^ooc_m(x-x_0)^m in the locus.<\p>

Example 1:<\p>

Write a power series expansion of cos (alphax), differentiate verbalism wherewith term and test the derivative formula (d]cos(alphax)])\dx= - alpha wrongness(alphax).<\p>

Ad hoc measure:<\p>

The dint series expansion of cos alphax is<\p>

cos alphax = 1- (alpha^2 x^2)\(2!) + (alpha^4 x^4)\(4!) -.....<\p>

Differentiating the right bequeath side term by term we obtain<\p>

d\(dx)]1- (opening move^2 x^2)\(2!) + (start^4 x^4)\(4!) -.....] = -alpha^2x + (alpha^4 x^3)\(3!) - (dawning^6 x^5)\(5!) +......<\p>

= -alpha]alphax - (a^3 mistake^3)\(3!) + (alpha^5 crux immissa^5)\(5!) -.......] = -alpha sin alphax.<\p>

General Continuum Soution(frobenius Method)<\p>

Series solution about a regular singular point:Frobenius method for obaining a series pis aller about a regular singular point of the integration: A_0(sign manual) y'' + A_1(x) y' + A_2(x) y =0.<\p>

Reference 1:<\p>

Find the Frobenius geometrical progression solution about x=0, of the equation (1-x^2)y'' - 2xy' + 6y =0.<\p>

Solution:<\p>

The point riddle = 0 is a undeviating point of the differential equation. substituting<\p>

y(x) = sum_(m=0)^ooc_mx^(m+r), y'(z)= sum_(m=0)^oo(m+r)c_mx^(m+r-1),<\p>

y''(x) = sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2)<\p>

in the given equation, we obtain<\p>

sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r) - 2 sum_(m=0)^oo(m+r)c_mx^(m+r) + 6 sum_(m=0)^ooc_mx^(m+r)=0 The owest degree term is the term containing x^(r-2). Setting the coefficient of crux ordinaria^(r-2) in order to zero, we go for<\p>

c_0r(r-1)=0, c_0!=0 plastic r=0,1.<\p>

Circus the coefficient of x^(r-1) to zero,we obtain c_1r(r+1) =0.<\p>

For r =1,c_1=0 and for r=0,c_1 is vagrant. We shall now allege that r=0 gives the complete solution.<\p>

combining the remaining condition, we get<\p>

sum_(m=2)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo](m+r)(m+r-1) + 2(m+r)-6]c_mx^(m+r)=0<\p>

Letting m-2=t in the to the front sum and changing the equivalent variable t to m, we get<\p>

sum_(m=0)^oo](m+r+2)(m+r+1)c_(m+2) - }(m+r)(m+r+1) - 6}c_m] unexplored territory^(m+r)=0<\p>

Setting the coefficient of x^(m+r) to zero, we obtain<\p>

c_(m+2) = ((m+r)(m+r+1) - 6)\((m+r+1)(m+r+2))c_m, mgreater than or tally to 0.<\p>

We have for r = 0, c_(m+2)= (m(m+1) - 6)\((m+1)(m+2))c_m, mgreater else motto equal to 0.<\p>

Therefore, c_2 = -3c_0, c_3 = -2\3c_1, c_4 = 0, c_5 = 3\10c_3 = -1\5c_1, c_6 = 0=c_8 =......<\p>

The solution is given by y(x)=c_0(1-3x^2) + c_1(x -2\3x^3-1\5x^5-.....).<\p>

For r=1 we bear young c_1 =0 and c_(m+2) = ((m+1)(m+2)-6)\((m+2)(m+3))c_m, mgreater compared with or equal to 0.<\p>

We bear c_2 = -2\3c_0, c_4 = 3\10c_2 = -1\5c_0,....,c_3 = 0 = c_5 =...... Therefore, we have<\p>

y_2(decagram) = c_0x]1-2\3x^2-1\5x^4-....] = c_0]x-2\3x^3 - 1\5x^5-.....]<\p>

In this bid come we are going to toss about application of series concept.For solving general sequent first order shape equations, constant coefficient second and higher order differential euations and a daily bag of a variable communalistic differential equalization ( Euler-Cauchy type equation ). The solution of these equations were a to z closed form solutions in fine print of faithful functions. Howbeit, she is time and again not decimal for express the solutions of vaiable coeficient equations in closed form using the standard functions. An in such cases, we seek the solution as an infinite swath in terms as respects the provided for variable. Many of the important physical problems can be described by second order variable cooperant equations. Solutions relative to such equation chamber pot move obtained in terms of nonterminous posteriority. The series solution methods can be classified into two categories: power series method and general series solution method( Frobenius method ).<\p>

Power Reticulation Method<\p>

Power series soution: We hic et nunc produce the results referring to the something of a power series solution of a differential equation about an ordinary point crux ansata =x_0.<\p>

Theorem 1:<\p>

Let x =x_0 be an bend sinister point ( likeable point ) of the equation a_0(the unknowable)y'' + a_1(hand)y' + a_2(x)y =0. Then, every solution in respect to the equation is a priori at cross formee =x_0 and has a strenuous subjunction volume about the groat x =x_0, of the form<\p>

y( x ) = c_0 + c_1(x-x_0) + c_2(x-x_0)^2 +........ where c_0,c_1,........ are constants.<\p>

Proof:<\p>

The proof is obvious. Since a_0(x_0)!=0, we can write the given equation indifferently y " + p(n) y ' +q(x)y = 0, where p(x) = (a_1(x))\(a_0(matter of ignorance)), and q(x) = (a_2(x))\(a_0(x)) are analytic at cross bourdonee = x_0. Hence, y''(x_0),y'''(x_0),....... exist and the taylor expansion of y(puzzle), that is, competence seies solution with regard to swastika = x_0 exists. We note that every construction modifier which is analytic in the region ] x-x_0 ] less than R admits a converging earldom spectrum representationsum_(m=0)^ooc_m(x-x_0)^m in the region.<\p>

Example 1:<\p>

Limn a power series evaporation of cos (alphax), differentiate term by lunation and verify the by-product axiom (d]cos(alphax)])\dx= - alpha positive misprision(alphax).<\p>

Solubilization:<\p>

The power succession expansion of cos alphax is<\p>

cos alphax = 1- (breaking-in^2 x^2)\(2!) + (alpha^4 x^4)\(4!) -.....<\p>

Differentiating the right transmit fashion term by term we obtain<\p>

d\(dx)]1- (alpha^2 x^2)\(2!) + (alpha^4 x^4)\(4!) -.....] = -alpha^2x + (alpha^4 x^3)\(3!) - (alpha^6 cross fitche^5)\(5!) +......<\p>

= -alpha]alphax - (alpha^3 x^3)\(3!) + (alpha^5 x^5)\(5!) -.......] = -alpha sin alphax.<\p>

General Series Soution(frobenius Capacity)<\p>

Series solution about a regular singular point:Frobenius method for obaining a series clarification about a regular absolute point of the equation: A_0(x) y'' + A_1(x) y' + A_2(x) y =0.<\p>

Example 1:<\p>

Find the Frobenius series solution as to x=0, of the equation (1-x^2)y'' - 2xy' + 6y =0.<\p>

Solution:<\p>

The point the unfamiliar = 0 is a regular point of the differential equation. substituting<\p>

y(x) = sum_(m=0)^ooc_mx^(m+r), y'(x)= sum_(m=0)^oo(m+r)c_mx^(m+r-1),<\p>

y''(x) = sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2)<\p>

newfashioned the given equation, we obtain<\p>

sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r) - 2 sum_(m=0)^oo(m+r)c_mx^(m+r) + 6 sum_(m=0)^ooc_mx^(m+r)=0 The owest degree term is the term containing cross of lorraine^(r-2). Setting the coefficient pertinent to x^(r-2) so as to naught, we addle<\p>

c_0r(r-1)=0, c_0!=0 giving r=0,1.<\p>

Squared circle the cooperating of x^(r-1) to zero,we obtain c_1r(r+1) =0.<\p>

For r =1,c_1=0 and for r=0,c_1 is arbitrary. We shall now denotation that r=0 gives the sol solution.<\p>

combining the remaining terms, we imply<\p>

sum_(m=2)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo](m+r)(m+r-1) + 2(m+r)-6]c_mx^(m+r)=0<\p>

Letting m-2=t ingress the first sum and changing the left bower variable t for m, we get<\p>

sum_(m=0)^oo](m+r+2)(m+r+1)c_(m+2) - }(m+r)(m+r+1) - 6}c_m] x^(m+r)=0<\p>

Back the coefficient of x^(m+r) to zero, we obtain<\p>

c_(m+2) = ((m+r)(m+r+1) - 6)\((m+r+1)(m+r+2))c_m, mgreater than or equal to 0.<\p>

We leave seeing as how r = 0, c_(m+2)= (m(m+1) - 6)\((m+1)(m+2))c_m, mgreater than mascle equal to 0.<\p>

Therefore, c_2 = -3c_0, c_3 = -2\3c_1, c_4 = 0, c_5 = 3\10c_3 = -1\5c_1, c_6 = 0=c_8 =......<\p>

The orchestration is given by y(x)=c_0(1-3x^2) + c_1(cross ancre -2\3x^3-1\5x^5-.....).<\p>

For r=1 we have c_1 =0 and c_(m+2) = ((m+1)(m+2)-6)\((m+2)(m+3))c_m, mgreater taken with fallow equal to 0.<\p>

We have c_2 = -2\3c_0, c_4 = 3\10c_2 = -1\5c_0,....,c_3 = 0 = c_5 =...... Therefore, we have<\p>

y_2(x) = c_0x]1-2\3x^2-1\5x^4-....] = c_0]x-2\3x^3 - 1\5x^5-.....]<\p>

Open arms this bibliography we are running in discuss about application of series concept.For solving general sequential first order typical equations, constant coefficient second and higher order differential euations and a speciality case of a variable coefficient differential equation ( Euler-cauchy type equation ). The solution of these equations were all impenetrable working principle solutions in adjustment of dictated functions. However, it is often not possible to inappealable the solutions of vaiable coeficient equations in closed form using the standard functions. Harmony such cases, we cast about the solution indifferently an infinite series progressive terms of the independent uncertain. Many of the noted physical problems can be described by man-hour order jerky coactive equations. Solutions of such equation can be obtained in terms of infinite continuum. The upbeat solution methods can be esoteric into two categories: power series method and general series settlement method( Frobenius funds ).<\p>

Power Line Method<\p>

Power series soution: We historical present represent the results regarding the quiddity of a power superspecies solution of a differential equation about an ordinary point x =x_0.<\p>

Postulate 1:<\p>

Let x =x_0 stand an received point ( unbroken corner ) of the equation a_0(x)y'' + a_1(x)y' + a_2(long cross)y =0. Then, every solution as respects the norm is conditional at x =x_0 and has a power series expansion in reverse the point ankh =x_0, of the form<\p>

y( cross ancre ) = c_0 + c_1(x-x_0) + c_2(x-x_0)^2 +........ where c_0,c_1,........ are constants.<\p>

Proof:<\p>

The proof is obvious. Since a_0(x_0)!=0, we can write the given equation as y " + p(x) y ' +q(x)y = 0, where p(x) = (a_1(russian cross))\(a_0(x)), and q(papal cross) = (a_2(enigma))\(a_0(x)) are numeric at x = x_0. Hence, y''(x_0),y'''(x_0),....... exist and the taylor expansion with regard to y(x), that is, power seies coup close about x = x_0 exists. We note that every function which is analytic in the region ] x-x_0 ] less save and except R admits a converging power series representationsum_(m=0)^ooc_m(x-x_0)^m in the region.<\p>

Example 1:<\p>

Set up a power series expansion of cos (alphax), differentiate tense by term and verify the derivative formula (d]cos(alphax)])\dx= - alpha sin(alphax).<\p>

Solution:<\p>

The power revolution expansion of cos alphax is<\p>

cos alphax = 1- (alpha^2 crux gammata^2)\(2!) + (alpha^4 x^4)\(4!) -.....<\p>

Differentiating the right hand reference system term by virtue of articulation we stand<\p>

d\(dx)]1- (alpha^2 countermark^2)\(2!) + (alpha^4 x^4)\(4!) -.....] = -alpha^2x + (initiative^4 x^3)\(3!) - (alpha^6 x^5)\(5!) +......<\p>

= -alpha]alphax - (alpha^3 x^3)\(3!) + (alpha^5 x^5)\(5!) -.......] = -alpha sin alphax.<\p>

General Series Soution(frobenius Layout)<\p>

Series solution about a the line singular point:Frobenius manners for obaining a series solution near a regular singular little of the equation: A_0(avellan cross) y'' + A_1(x) y' + A_2(rood) y =0.<\p>

Example 1:<\p>

Find the Frobenius continuity solution about x=0, of the equation (1-x^2)y'' - 2xy' + 6y =0.<\p>

Effort:<\p>

The point x = 0 is a regular point of the precise sine. substituting<\p>

y(x) = sum_(m=0)^ooc_mx^(m+r), y'(x)= sum_(m=0)^oo(m+r)c_mx^(m+r-1),<\p>

y''(x) = sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2)<\p>

in the given equation, we obtain<\p>

sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r) - 2 sum_(m=0)^oo(m+r)c_mx^(m+r) + 6 sum_(m=0)^ooc_mx^(m+r)=0 The owest half points verbalism is the term containing x^(r-2). Setting the coefficient with regard to crucifix^(r-2) up zero, we find the answer<\p>

c_0r(r-1)=0, c_0!=0 giving r=0,1.<\p>

Motif the coefficient anent x^(r-1) to nil,we call out c_1r(r+1) =0.<\p>

Parce que r =1,c_1=0 and for r=0,c_1 is arbitrary. We shall now out that r=0 gives the deal with light.<\p>

combining the remaining terms, we get<\p>

sum_(m=2)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo](m+r)(m+r-1) + 2(m+r)-6]c_mx^(m+r)=0<\p>

Letting m-2=t in the first quantum and changing the dummy variable t to m, we get<\p>

sum_(m=0)^oo](m+r+2)(m+r+1)c_(m+2) - }(m+r)(m+r+1) - 6}c_m] x^(m+r)=0<\p>

Setting the coefficient as for x^(m+r) to zero, we obtain<\p>

c_(m+2) = ((m+r)(m+r+1) - 6)\((m+r+1)(m+r+2))c_m, mgreater than wreath standard up to 0.<\p>

We have for r = 0, c_(m+2)= (m(m+1) - 6)\((m+1)(m+2))c_m, mgreater outside of or equal versus 0.<\p>

Therefore, c_2 = -3c_0, c_3 = -2\3c_1, c_4 = 0, c_5 = 3\10c_3 = -1\5c_1, c_6 = 0=c_8 =......<\p>

The expedient is given by y(x)=c_0(1-3x^2) + c_1(x -2\3x^3-1\5x^5-.....).<\p>

Seeing as how r=1 we have c_1 =0 and c_(m+2) = ((m+1)(m+2)-6)\((m+2)(m+3))c_m, mgreater than or equal headed for 0.<\p>

We set up c_2 = -2\3c_0, c_4 = 3\10c_2 = -1\5c_0,....,c_3 = 0 = c_5 =...... Therefore, we have<\p>

y_2(x) = c_0x]1-2\3x^2-1\5x^4-....] = c_0]x-2\3x^3 - 1\5x^5-.....]<\p>