Power series solutions | Series Solutions of Differential Equations

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Power series solution of differential equations

Series Solution of Linear Ordinary Differential Equations

Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
In particular, we shall obtain
• the form of the series expansion,
• a recurrence relation for determining the coefficients

Given the differential equation \(\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0\), substitute\(y=\sum_{n=0}^{\infty}{a_n(x − x_0)^n }\) and solve for the \(a_n\) to find a power series solution centered at \(x_0\).

Power Series
A series of the form \(\sum_{n=0}^{\infty}{a_n(x − x_0)^n } =a_0 + a_1(x − x_0) + a_2(x − x_0)^2 + · · · \) is called a power series about the point \(x_0\). Here, x is a variable and \(a_n\)’s are constants.

Analytic function
A function \(f(x)\) defined on an interval containing the point \(x_0\) is called analytic at \(x_0\) if its Taylor series about \(x_0\), i.e., \(\sum_{n=0}^{\infty}{\frac{f^n(x_0)}{n!}(x − x_0)^n }\) exists and converge to \(f(x)\) for all x in some nbd of \(x_0\).
For example, \(e^x, sinx, cosx, sinhx\) are analytic everywhere, where as the function \(\frac{x-2}{x-3}\) is analytic except \(x=3\).

Power Series Solutions to D.E.s at Ordinary Point

Let us consider a differential equation in the standard form :\(\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0\)

Definition

A point \(x_0\) is an ordinary point if both P(x) and Q(x) are analytic at \(x_0\). If a point in not ordinary it is a singular point.

Solved Problems on Power series method about an ordinary point

Prob.(1): Find the ordinary and singular points of the D.E. \((1-x^2)\frac{d^2y}{dx^2} -2x\frac{dy}{dx} + 2y = 0\)

Prob.(2): Consider the differential equation \((x-1)\frac{d^2y}{dx^2} +x\frac{dy}{dx} + \frac{1}{x} y = 0\)
Then
1. x=1 is the only singular point
2. x=0 is the only singular point
3. both x=0 and x=1 are singular points
4. neither x=0 nor x=1 are singular
points
[NET 2017(D)]

Prob.(3): Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} +3x\frac{dy}{dx} + 3y = 0\) about x=0
[C.H. 2006] [C.H. 2009] [C.H. 2012] [C.H. 2015]

EXERCISE

  1. Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} +3x\frac{dy}{dx} + 3y = 0\) near the ordinary point x=0
    [C.H. 2006] [C.H. 2009] [C.H. 2012] [C.H. 2015]
  2. Find the power series solution of  the D.E. \((1-x^2)\frac{d^2y}{dx^2} +2x\frac{dy}{dx} -y = 0\) about x=0
    [C.H. 2011] [C.H. 2013
  3. Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} – y = x\) about x=0
    [C.H. 2007] [C.H. 2009
  4. Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} +x\frac{dy}{dx} +x^2 y = 0\) about x=0
    [C.H. 2013] [C.H. 2014]
  5. Find the power series solution in power of x:\(\frac{d^2y}{dx^2} + y = 0\) near the ordinary point x=0
    [C.H. 2007
  6. Find the power series solution of  the D.E. \((x^2+1)\frac{d^2y}{dx^2} +x\frac{dy}{dx} -x y = 0\) about x=0
    [C.H. 2014
  7. Find the power series solution of  the D.E. \((1-x^2)\frac{d^2y}{dx^2} +2x\frac{dy}{dx} – y = 0\) about x=0
  8. Find the power series solution of  the D.E. \((x^3-1)\frac{d^2y}{dx^2} +x^2\frac{dy}{dx} +xy = 0\) about x=0 
  9. Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} -4(x-1)y = 0\) near the ordinary point x=1
    [C.H. 2008
  10. Find the power series solution of  the D.E. \(\frac{d^2y}{dx^2} +(x-1)^2\frac{dy}{dx} -4(x-1) y = 0\) near the ordinary point x=1
    [C.H. 2012
  11. Find the power series solution of the following initial-value problems:
    (i) \(\frac{d^2y}{dx^2} -4y = 0\) satisfying \(y(0)=1, y^{/} (0)=-1\)
    [C.H. 2008] [C.H. 2011]
    (ii) \(\frac{d^2y}{dx^2} -x\frac{dy}{dx}-y = 0\) satisfying \(y(0)=1 , y^{/} (0)=0\)
    [C.H. 2009]
    (iii) \(\frac{d^2y}{dx^2} +x\frac{dy}{dx}-2y = 0\) satisfying \(y(0)=1 , y^{/} (0)=1\)
    [C.H. 2009]
    (iv)\(\frac{d^2y}{dx^2} +x\frac{dy}{dx}+2y = 0\) satisfying \(y(0)=1 , y^{/} (0)=1\)
    [C.H. 2010]
    (v) \(\frac{d^2y}{dx^2} +8x\frac{dy}{dx}-4y = 0\) satisfying \(y(0)=1 , y^{/} (0)=0\)
    [C.H. 2010]
  12. Find the power series solutions \(x^2\frac{d^2y}{dx^2} +3x\frac{dy}{dx}-y = 0\)
  1. \( y(x)=a_0 (1-\frac{3}{2} x^2+\frac{9}{8} x^4-\dots)+a_1 (x-x^3+\frac{3}{5} x^5-\dots)\)
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  3. \(y(x)=a_0 (1+\frac{1}{2} x^2+\frac{1}{24} x^4+⋯)+a_1 (x+\frac{1}{6} x^3+\frac{1}{120} x^5+⋯)+\\(\frac{1}{6} x^3+\frac{1}{120} x^5+⋯)\)