Impetration relative to Series
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>
Even this solution is the industrious multiple of the second solution newfashioned equation (1). The atomic points of the reciprocal are x = +- 1 and the series waxing is holographic about x=0. Therefore, the radius of convergence is R = 1.<\p>