Solving the Second Order Linear Differential Equation using the Power Series Method

 The Power Series Method requires no introduction. It is one of the most powerful methods of analyzing functions or solving differential equations. Solving this differential equation using the power series will circumnavigate the use of complex numbers thus providing a much more ''physical" understanding of the concepts involved. They also provide insight about an important and alternate definition of the sine and the cosine functions, the so called "Taylor Series" expansion of the sine and the cosine functions. They show how sinusoids are related to infinite series also. So, the overall impression of the Power Series Method is that it is a versatile and a powerful method for solving differential equations. 

We always strive to explain nature and its phenomena in a deterministic and a well defined, logical manner. We use mathematics as the guardrail of our understanding to achieve this endeavor. To explain the various natural laws mathematically, differential equation arises and so we have developed various methods to solve the same. One the most interesting phenomena that has received ample well deserved mathematical attention is oscillation, more precisely, sinusoidal oscillations. We shall see how they arise as a consequence of simple physical laws and shall use one of the methods to study them.

The most basic form of oscillation, the Simple Harmonic Motion arises from simple systems such as a spring and a mass on a frictionless suitable support. The most empirical law dealing with such systems is the Hooke's Law which states that the restoring force exerted by a stretched spring is directly proportional to the elongation or the change in length of the spring.

Expressing this law mathematically-

where 'm' is the mass, 'x' is the position of the mass, 'k' is the spring constant and 't' is the time. The left-hand side of the expression originates from Newton's 2nd Law, the right-hand side from Hooke's Law. The negative sign indicates that the force is restoring in nature.

Now, to solve this using the Power Series method, we express the position 'x' as an infinite series of polynomials with constant coefficients (Power Series) further, 'x' is a function of time. This a crucial step.

In summation notation-

Now, differentiating 'x' w.r.t 't', we get-

Differentiating once again-

The dot sign in the expression indicates multiplication. Due to absence of space, the remaining terms had to written in another row i.e. the second row is the continuation of the first one.

The summations are getting pretty huge ;)

That's all the differentiation we had to do. Now, we equate the left and the right sides of the equation.
                                         

Equating corresponding coefficients-

Expressing 'x' ,which is the original series ,with the derived coefficients-

Taking the coefficients common-

Therefore 'x' can be taken as a linear combination of two different functions. Now, these two functions are quite remarkable. Graphing them clears the mystery. The two functions can be expressed as an infinite summation of a series of polynomials.

 The upper series is the Taylor Series Expansion of the Cosine Function. The lower one is the Expansion of the Sine Function. Thus Trigonometric Functions can be expressed as an infinite summation of polynomials with appropriate coefficients. This also shows the oscillatory nature of the solution to the Differential Equation being solved.

Thus, the actual general solution of the differential equation is-

The particular solution to this differential equation depends on the initial conditions, for example, the initial position and the initial velocity of the mass etc. The initial conditions helps us find the value of the independent constants in the general solution, thus completing the solution. This also shows that a Second Order Linear Differential Equation has a solution that is the linear combination of two solutions.

We have thus come to the end of this article. This article shows just how 'powerful' Power Series can be. The power series solution not only solves the differential equation but also yields the Taylor Series Expansion of the Sine and the Cosine function in the process, giving us further insight about their nature.

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