# 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

- 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**] - 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**] - 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**] - 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**] - 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**] - 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**] - 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
- 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
- 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**] - 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**] - 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**] - Find the power series solutions \(x^2\frac{d^2y}{dx^2} +3x\frac{dy}{dx}-y = 0\)

- \( y(x)=a_0 (1-\frac{3}{2} x^2+\frac{9}{8} x^4-\dots)+a_1 (x-x^3+\frac{3}{5} x^5-\dots)\)
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- \(y(x)=a_0 (1+\frac{1}{2} x^2+\frac{1}{24} x^4+⋯)+a_1 (x+\frac{1}{6} x^3+\frac{1}{120} x^5+⋯)+\\(\frac{1}{6} x^3+\frac{1}{120} x^5+⋯)\)