Solution to Ordinary Differential Equations Assignment
During your first year, you studied calculus and did the different types of differentiation methods, integration methods, and thenormal equations. However, when you advance to the second year and third year, things start getting messier. Till now, you were accustomed to providing a solution to an equation, but here you will find other types of equations whose solution is a function and the equation that you are solving contains derivatives. Tricky, isn’t it? It will require you to focus a lot in class and do a lot of practice on your own. When an equation is made up of a derivative, and the solution is a function, it is known as an ordinary differential equation. Sometimes it is abbreviated as ODE. It is a unit on its own in most campuses, and assignments are generally not as easy as most would expect. Do not worry. Hereby, Iam suggesting you the best online platform that you can visit to get quality ordinary differential equations assignment help.
Ordinary differential equation
We have explained to you what an ordinary differential equation should be like. Here is an example of a simple ordinary differential equation
The above equation has a derivative on the right-hand side. Thus we say it is an ODE. To solve it we have to use an antiderivative. Using the trigonometric rules or checking from the SMP table, the antiderivative for cos(y) is sin(y). The function here is x(y). The solution of the equation is
X(y) = sin(y) +C
You might ask, why do we add a constant C? This value is added when there is an integration; it is an arbitrary function that must be added in any integration question that you solve. Some lecturers are strict about it and could deduct marks for the failure of adding the arbitrary constant. To be on the safe side, ensure that you always add it to the equation. From the above equation, it can be written as X’. The apostrophe signifies that it is a derivative. This is an equation involving a derivative that confirms to you that it’s an ODE. In our example, it’s on the left-hand side. It could also be on the right-hand side and could be more than one. If there are more than one derivatives in the equation, the process of finding the solution could become very complicated. That is when it’s wise to seek help.
Order of a differential equation.
Like in the polynomial equations, there is an order of an ordinary differential equation. An order of an equation is the highest derivative in the equation. For the above case, it had an order of one. The order of an equation can increase to two, 3, 4, up to 10. There are no restrictions.
Linear differential equation.
This is a differential equation of the first order. It involves the function of y and its derivatives and has lots of applications in real life. It can contain two continuous functions and a derivative. If either of the continuous functions is equal to zero, the equation can be reduced in such a way that it will be easier to solve.
Why are differential equations useful?
Differential equations are taught in many courses in universities. Examples of such courses are economics, mathematics, biology, chemistry, and engineering. If it’s being taught in most of the universities, then it has lots of applications. How useful is it in real life?
ODE has been used in one of the population models known as Malthusian law named after the person who developed it, Malthus. He used this law to predict how the population of the world would grow. It can also be used to study other population issues.
Any electrical equipment that you use is an outcome of the differential equation.
Different methods for solving ODE
- Euler’s method.
This method offers another alternative to the algebraic methods of solving the ordinary differential equation. It is important to note that not all-real life application of ODE can be solved algebraically. A numerical approach also gives a good approximation of the values. Euler’s method makes use of the Taylor series to solve an equation. The resulting Taylor series will have many terms. The accuracy of the method is enhanced if more terms are used.
- Runge-Kutta method.
This is another numerical method of solving an ordinary differential equation. This method is named after its developers (Wilhelm Kutta and Carl Runge), who developed it in 1900. It’s used in different fields because of its simplicity. It is also a very reliable predictor.
The hamming method was developed in 1959 by Richard Wesley Hamming from whom it derives its name. It’s a powerful predictor method for uncertain Ordinary differential equations.
From the numerical analysis, this method has shown to be more effective and efficient than the above methods that we have listed above.
This method only solves the boundary value problems. We can say that we ‘shoot’ to the different trajectories until we get the best boundary value. For an ordinary differential equation, we assume an initial value problem if the ODE was an initial value. We then proceed on to finding the boundary values; the ones obtained are compared with the true boundary values. To get to the best boundary value, the method involves trial and error.
Solving ODE in Matlab.
Matlab is a programming language that has gained a lot of scientific community support due to its versatility. With the software, you can solve any mathematical problem with ease provided that you know how to use it. In addition, you won’t have the challenge of installing other packages to solve the problem. It has readymade functions that you can use. The basic solvers that you can use to solve an ODE problem are ode45, ode23, and ode113. With Matlab, you can solve the different types of ordinary differential equations, such as linear differential equations and explicit differential equations.
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