Introduction to Second Order Differential Equations
Second-order differential equations involve second derivatives and can be categorised into two main types: homogeneous and non-homogeneous. These equations often appear in physics, engineering, and other scientific disciplines to describe systems where the acceleration or the second derivative of a quantity with respect to time is a function of displacement, velocity, and time. Understanding the basics of differentiation is crucial before diving into second-order differential equations.
General Form
The general form of a second-order linear differential equation is:
d2y/dt2 + p(t)dy/dt + q(t)y = g(t)
Where:
- p(t), q(t), g(t) are given functions,
- y is the unknown function to be determined.
Homogeneous Second Order Differential Equations
Definition and General Solution
Homogeneous second-order differential equations are those where g(t) = 0 in the general form. The general solution of a homogeneous equation is the sum of the complementary function and the particular integral.
Solving Homogeneous Equations
To solve homogeneous equations, we assume a solution of the form y = e(rt), where r is a constant to be determined by substituting the assumed solution into the equation and solving the resulting algebraic equation. Familiarity with the differentiation of trigonometric functions and differentiation of exponential and logarithmic functions can provide further insight into solving these equations.
Example 1: Simple Harmonic Motion
Consider a mass-spring system described by the differential equation:
d2y/dt2 + k/m y = 0
Solution: Assuming y = e(rt) and substituting into the equation, we find the characteristic equation r2 + k/m = 0. Solving for r gives the solutions required to write the general solution.
Non-Homogeneous Second Order Differential Equations
Definition and General Solution
Non-homogeneous second-order differential equations have g(t) ≠ 0. The general solution is typically obtained by finding a particular solution of the non-homogeneous equation and adding it to the general solution of the associated homogeneous equation. This process often involves the techniques of integration, which are essential for finding particular solutions.
Methods of Solution
- Method of Undetermined Coefficients: Useful when g(t) is a simple function (e.g., polynomial, exponential, or sine/cosine).
- Variation of Parameters: Applicable when the method of undetermined coefficients is not suitable.
Example 2: Electrical Circuit
Consider an electrical circuit described by the equation:
Ld2q/dt2 + Rdq/dt + (1/C)q = E(t)
Where:
- L is inductance,
- R is resistance,
- C is capacitance,
- E(t) is an electromotive force.
Solution: A particular solution to the non-homogeneous equation is found, and it is added to the general solution of the associated homogeneous equation to obtain the general solution.
Applications and Examples
Mechanical Vibrations
Second-order differential equations describe mechanical vibrations, such as a mass-spring-damper system, where the equation represents the balance of forces acting on the system. The study of first-order differential equations can also provide foundational knowledge for understanding these systems.
Electrical Circuits
In electrical circuits, these equations describe the voltage and current relationships among the elements (resistors, capacitors, and inductors).
Heat Conduction
In heat conduction, they model the distribution of heat in a solid over time.
Example Questions within Notes
Example Question 1: Given the differential equation d2y/dt2 - 3dy/dt - 4y = 0, find the general solution.
Solution: Assuming y = e(rt), substituting, and solving the characteristic equation r2 - 3r - 4 = 0, we find r = 4, -1. Thus, the general solution is y(t) = c1e(4t) + c2e(-t), where c1 and c2 are arbitrary constants.
Example Question 2: Solve the non-homogeneous differential equation d2y/dt2 + 2dy/dt + y = 3e(-t).
Solution: First, solve the homogeneous equation to find the complementary function. Then, find a particular integral and add it to the complementary function to find the general solution.
FAQ
Absolutely, second-order differential equations find applications across various fields beyond mechanical and electrical systems. In biology, they can model populations or the spread of diseases. In finance, they can describe the evolution of prices in financial markets. In physics, they can represent wave propagation, heat distribution, or quantum mechanical phenomena. Essentially, whenever a quantity’s rate of change is influenced by the quantity itself and its rate of change, second-order differential equations can be a potent tool for modelling and analysing such systems.
The method of undetermined coefficients is typically used when the non-homogeneous term g(t) is a simple function like a polynomial, exponential, or a sine/cosine function, and especially when it is similar to the complementary function. If the method of undetermined coefficients is not suitable, typically when g(t) is not of a simple form or is a product of several functions, the variation of parameters method is employed. Variation of parameters involves finding a particular solution by assuming its form involves undetermined functions (instead of coefficients) and using the original differential equation to determine these functions.
The characteristic equation is pivotal in solving second-order linear homogeneous differential equations as it provides the exponents in the exponential solutions of the equation. When we assume a solution of the form y = e(rt) and substitute it into the differential equation, we obtain an algebraic equation in terms of r, known as the characteristic equation. The solutions to this equation, r1 and r2, provide the general solution to the differential equation as y(t) = c1e(r1t) + c2e(r2t), where c1 and c2 are arbitrary constants. The characteristic equation thus plays a crucial role in determining the behaviour of the solutions.
When the characteristic equation of a second-order linear homogeneous differential equation has complex roots, the solution takes on a different form. If the roots are complex conjugates, say r = a ± bi, where a and b are real numbers, the general solution of the differential equation is given by y(t) = e(at)(c1 cos(bt) + c2 sin(bt)), where c1 and c2 are arbitrary constants. The presence of complex roots introduces oscillatory behaviour to the solution, represented by the sine and cosine terms, and e(at) represents the damping or growth of the oscillations depending on the sign of a.
The method of undetermined coefficients is a technique to find a particular solution yp of a non-homogeneous second-order linear differential equation. The method is applicable when the non-homogeneous term g(t) is of a form that will yield to the method, such as a polynomial, exponential, sine, or cosine function. The idea is to guess a form for yp that is similar to g(t), but with undetermined coefficients. These coefficients are then determined by substituting yp and its derivatives into the differential equation and equating the resulting expression with g(t). The coefficients are chosen to satisfy the equation identically for all t.
Practice Questions
The characteristic equation associated with the differential equation d2y/dt2 - 5dy/dt + 6y = 0 is r2 - 5r + 6 = 0. Factoring this, we get (r - 2)(r - 3) = 0, so r = 2, 3. Therefore, the general solution of the differential equation is y(t) = c1e(2t) + c2e(3t), where c1 and c2 are arbitrary constants determined by initial conditions, which are not provided in this case. This solution represents a superposition of two exponential functions, each corresponding to a solution of the characteristic equation.
To find the particular solution, we first find the complementary function by solving the associated homogeneous equation d2y/dt2 + 3dy/dt + 2y = 0. The characteristic equation is r2 + 3r + 2 = 0, which factors as (r + 1)(r + 2) = 0, giving r = -1, -2. So, the complementary function is yc(t) = c1e(-t) + c2e(-2t). To find a particular solution yp(t) to the non-homogeneous equation, we try yp(t) = Ae(-t), substituting it and its derivatives into the non-homogeneous equation to find A. Adding yc and yp gives the general solution, and applying the initial conditions allows us to find c1 and c2.
