Find the Differential Equation of the Family of Parabolas with vertex and focus on the x-axis

Given the equation of a parabola

  

We let (h,0)  be the location of the vertex, and  (h+p,0)be the location of the focus where will be any number along the x-axis and p is the distance of the focus added to the vertex’s distance h. This is to represent the distance traveled by the focus from the point of origin.

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The curves are then sideways parabolas with equation

Now we need to find a differential equation that does not depend on arbitrary constants  and p. Since there are two arbitrary constants to deal with, we need to find its second differentiation. In other words, the highest order derivative that should appear in the differential equation is y”, i.e. the second derivative of y with respect to x.

Now let us differentiate the equation using implicit differentiation.

That gives

which still depends on the constant p.

Since there is still an arbitrary constant which is p , we need to get its second derivative using implicit differentiation for the second time

This last equation does not depend on the constants h  and p.
The desired differential equation is

 

The Importance of Differential Equations

We live in the age of astonishing advancement. Engineers can create robots, physicist can describe the motion of waves, pendulums or chaotic systems, while we communicate wirelessly in a vast world wide network. But underline this modern wonders, are deep and mysteriously powerful, they are called Differential Equations. But what are differential equations, where does differential equations come from and why does they work so well in a wide variety of discipline such as biology, physics, chemistry, economics and even in engineering. Why are they not generally observed and used in our day to day life.

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Differential equations are equations that contains one or more terms involving derivatives of one variable (dependent variable) with respect to another variable (independent variable) or we can say that these are equations involving derivatives of a function or functions. They have a remarkable ability to predict the world around us. They are used to describe exponential growth and decay, population growth of species or the change in investment return over time, bank interest, even in solving radioactive decay problems, continuous compound interest problems, flow problems, cooling and heating problems, orthogonal trajectories, and also in investigating problems involving fluid mechanics, circuit design, heat transfer, population or conservation biology, seismic waves. They are used in specific field such as, in the field of medicine, where differential equations are used for modelling cancer growth or the spread of disease. In chemistry, they are used for modelling chemical reactions and to computer radioactive half-life. In economics, they are used to find optimum investments strategies. In physics, they are used to describe the motion of waves, pendulums or chaotic systems. They are also used in physics with Newton’s Second Law of Motion and the Law of Cooling that pertains to the temperature of objects and its surroundings. In engineering, they are used for describing the movements of electricity. Differential equations are also used in creating software to understand computer hardware belongs to applied physics or electrical engineering. They are also used in game features to model velocity of a character in games. They are essential tools for describing the nature of the physical universe and naturally also an essential part of models for computer graphics and vision. Differential equations are also used as aspect of algorithm on machine learning which includes computer vision. Also involves solving for optimal certain conditions or iterating towards a solution with techniques like gradient descent or expectation maximization. In Mother Nature, differential equations are essential tool for describing the nature of the physical universe. Even in networking, they are used to understand an outcome of an edge creation model like preferential attachment which says the nodes with probability proportional to their existing degrees. And also used in theories and explanations like using a determinants to estimate the area of the Bermuda triangle. Even in Bots (short for robots), partial and ordinary differential equations helps to provide shapes and interior and exterior designs of machine.

With their broad and advance uses, no wonder why some people found it difficult. However, with their remarkable ability, people spend their time analyzing and describing their behaviors. That’s why everyone of us should appreciate them.