Deferential equations – second order
Necessity is the mother of invention; it is the same with mathematics. A great discovery solves a great problem. Thinking mathematically could change and develop your life in many fields, including logical reasoning and problem-solving skills. Mathematics is vital in daily life, several types of employment, technology, medicine, economy, sciences environment and development in public decision-making. The earliest uses of mathematics included economic exchange, land allocation, drawing, and the recording of time, but now it is affecting the entire world. Study of mathematics satisfies a broad variety of interests and capabilities (Blanchard et al, 55).
It forms the imagination, teaches clearly and logically. Mathematics is challenging because it deals with complex ideas and difficult questions. Calculus is one of the most pertinent sections of the mathematics. Calculus deals with quantities, which are near other quantities with limits, so it is less static and more dynamic. Studying calculus course is a window of opportunity to other courses related to Mathematics. Learners will have a better understanding as they advance their studies. Today calculus has multiple uses including navigation in outer space, predicting population sizes, estimating how fast coffee prices rise, and expecting the weather (Argawal & Donal, 22).
Differential equation involves equations with not known function. The functions could belong to one or more variables relating the function’s value and derivatives. Differential equations are mainly used in engineering, economics and physics among other fields. Differential equations are applicable in many fields of science especially in situations of determining the relationship of quantities. These quantities are continuously varying, and their levels of changing are postulated or known. In mechanical engineering, the movement of a body is defined by position and velocity.
The formulas for differential equations differ depending on the form of equation. Studying differential equations is a broad field in pure mathematics as well as applied. Every discipline using differential equations focuses on one or more types of these equations. Differential equations are significant in creating biological, physical or technical processes. Some of these processes include making designs of structures and buildings, determining motion of celestial bodies and interaction of neurons among others. When using these equations in real life situations, one may not get a specific solution. In such circumstances, the answer is estimated using numerical formulae. For example, Bessel’s differential equation is:
This is a simplified form of a second order differential equation (Morris & Orley, 67).
A mass on a spring will decelerate because of friction, and the magnitude is proportional to velocity. The differential equation will be in this form:
C is the coefficient, which will be representing friction.
We could say m= 1 and 0 < c -2a and k= a2 + b2
This is a category with several equations of different aspects. Some of the equations include linear and nonlinear, reduction of order, Euler Cauchy, linear independence and Wronskian among others (Argawal & Donal, 78). The explicit form of second order equation is:
Second order linear equation could also have this form:
The solution can be solved in the following way:
Therefore, the solution for this equation is y = c1y1 + c2 y2
In this equation, P, Q, R, and G are direct forms of continuous functions. Such equations are mostly used in the study of spring motion. For example, a mass is fixed on a spring, and it exerts reasonable force, which is proportional to either extension or compression of the spring. All other forces are assumed to be constant. The equation will use Newton’s second law (Blanchard et al, 88).
M represents the mass on the spring and k stands for the measurement of the spring’s stiffness. Make m = k and use Euler’s theorem to get a solution. According to Euler’s theorem, the solution will take this format:
This means there is a need to find out the unknown constants, which are A & B. We need to specify the state of a given time. So t = 0, extension will be x = 1 and the particle will not be moving so it will be dx/dt =0. This will give the following equation:
And B = 0 hence x (t) = cos t.
Ordinary homogenous equation has this general form (Argawal, Ravi & Dona, 61):
The above equation can be solved in a closed format. This can happen by changing some variables. These variables are u = y/x and the equation will change to this form:
It is common to have homogenous equations with constant coefficients. Such an equation is expressed as
The general solution for this equation will be
A homogenous equation can also take characteristic form:
If the equation takes such a form, it is solved in the following criterion (Morris & Orley,75):
If the equation is homogenous, linear and in the second order category, it will take this form:
In this equation, a, b and c are constant. This equation has an auxiliary equation, which is
Solutions for the equations of this category are determined by the solution of the auxiliary equation. It begins with finding the roots of the auxiliary equation and they are given by the formulae of the quadratic equation (Morris & Orley, 107).
The roots may be in various forms, which will determine the general solution. The following are examples of possible roots.
Equations in this category take this form if it is free motion (Blanchard et al, 112).
They are mostly used in calculating motions of different aspects. They could be free motions or forced motions. Free motions are caused by gravity or the spring. If other external forces are involved, then it will be forced motion. Forced motion has this formula
Solving a Nonhomogenous equation is combining both homogenous and Nonhomogenous general solution. After they are combined, they form this solution
The final solution format will have such an equation:
Some equations will have coefficients, which are undetermined. For example, a student may be required to find the general solution for this equation:
The first step is solving the characteristic equation (Argawal & Donal, 122).
After getting these values, they should be substituted to the original equation (Morris & Orley, 132).
Another example is finding the general solution of
The characteristic equation is solved and the substitution (Morris & Orley, 111). The resulting equation will be:
The final general solution will be as follows:
Differential equations exist in different forms of categories. It is crucial to learn them since they are used in many fields. For one to understand integration, he or she will need to learn differential equations since they are dependent. They seem to have a long method of solving them, but practice enables learners to improve efficiency. Learners who would like to advance Mathematics studies will need knowledge in differential equations. These equations are vital to professionals who deal with processes requiring the use of these equations (Blanchard et al, 131).
Agarwal, Ravi P, and Donal O’Regan. Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems. New York: Springer, 2009. Print.
Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Pacific Grove, CA: Brooks/Cole Pub. Co, 2010. Print.
Morris, Max, and Orley E. Brown. Differential Equations. Englewood Cliffs, N.J: Prentice-Hall, 2009. Print.