Theory of Second-Order Constant-Coefficient Differential Equations

Posted: August 26th, 2021

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Theory of Second-Order Constant-Coefficient Differential Equations

Definition of Second-Order Constant-Coefficient Differential Equations

Second-Order Differential Equations is a theory about special hypergeometric functions that define solutions for the second-order linear differential equations. As such, a second-order differential equation refers to the differential equation that takes the following form;

…equation 1

In this case, a, b, c, and d are prescribed or constant-coefficients for x functions. Like the first order linear equation case, any interval with a(x) ≠0, it is possible to replace equation (1) using the standard form of the second-order linear equation as follows;

In which case; p (x) = b(x)/a(x), q(x) = c(x)/a(x) and g (x) = d(x)/a(x). Additionally, as revealed under equation 2, given that a, b and c are actual constants such that a ≠0, it then implies that

…equation 3 is defined as a constant coefficient equation (in this case, linear second-order constant-coefficient differential equation).Leonhard Euler first introduced differential equations with constant equations during a study of exponential functions denoted by. This is a unique solution for equation, whereby. Here, it implies that the nth derivative for  will be  which enables an easy solution for a homogenous linear equation.

Difference between Homogenous and Non-Homogenous Cases

The distinction between these two aspects of a differential equation is determined based whether either side of equation 2 is zero or non-zero;

.

Whenever either side is zero, then, the second-order differential equation is said to be homogenous and written as in the following form;

. However, if g (x) ≠ 0, then it is non-homogenous. Notable, however, is that the terminology, that is, homogenous is entirely not related to homogenous equations of degree zero.

Process for Finding General Solutions to Differential Equations

Solving Homogenous Equation

A second-order homogenous linear differential equation can be written as follows;

, whose polynomial characteristics are represented by

Therefore, given that a and b are actual operators, the solution for the equation can be achieved in three cases. However, this is determined by a discriminant. Accordingly, the general solution for the three cases is affected by two arbitrary constants denoted by c₁ and c₂. As such, the following conditions apply;

1. When D > 0, the polynomial contains two different actual roots, µ, and α. As such, the general solution for the equation would be;

1. However, when D = 0, then the polynomial will contain the double root, – µ/2. Hence, the general solution for the equation would be presented as;

1. Lastly, when D < 0, it implies that the polynomial equation contains two complex conjugate roots such that µ ± αi. Hence, the general solution would be;

The above solution can be rewritten as follows;

As such, the results y(x) that satisfied y (0) = d₁and. Here, the general solutions attained above are respectively equated to zero and their respective derivatives to  and. The resulting solution is a linear system for two linear equations having two constants (unknown) c1 and c2. The solution yields a Cauchy problem, where values for each component are specified at zero (0).

Solving Non-Homogenous Equation

A sample non-homogenous equation of order n having constant coefficient is written as;

, that is, In this equation, y1…ynis the basis for solutions of the vector space while µ1 … µn represents the arbitrary constants or unknown functions, which should be attained to find the solution for y in the non-homogenous equation. Hence, one should add constraints as follows;

, using induction and by-product rule, it implies that;

The equation and above equated to zero (0) on the left side represents an n linear equation system with known function coefficients as f, y_i, and their respective derivatives, which can be solved using linear algebra.

Illustration of Second-Order Constant-Coefficient Deferential Equation

Given, the equation is a homogenous one that can be solved as follows;

Let  hence this results to;

…..first order

….second order. Thus, these equations are substituted into the problem equation above resulting into;

, by simplifying the equation we get:

, thus the equation has been reduced to a differential equation to an ordinary quadratic equation that is specially referred to as characteristic equation, factored as follows;

, implying that r = 4, -3, resulting in the two solutions but not the final one. Since, yet this is not the final result. The answer is tested into the final equation to ensure that it yields to zero to both sides as proof of its validity.

Solution of an Initial Value Problem for a Non-Homogeneous Linear Second-Order Differential Equation

Given, find the resolution of the equation. An initial value solution can be obtained by solving related homogenous equation as follows;

However, corresponding roots would be;

. Thus, the general solution to the homogenous part of the equation is as follows;

.