Derivatives are an important section that you use in Calculus. Differential equations involve the dy/dx function that is widely used in Maths. The job of this differentiation is to calculate at what rate the value of that function will change at a fixed point. You can also get it with the tangent of the slope when you draw that graph. Now there are ways in which you can use more than one differentiation function together. When you combine them all, you get an entire equation called the differential equation. You can solve it with certain formulas you will learn in higher classes.
Definition and The Order
An equation can be generally defined as a mathematical term to represent a relationship between two or more things. When we say that x=y, we can easily come to the relationship that the two variables are equal. A differential equation is a different type of equation that includes the rate of change of things apart from the things and has a derivative as a term.
Let us for instance take a cup of hot water. So how long would it take to make the water reach room temperature? Some might answer that the water will lose heat according to the surrounding environment at a constant rate until it reaches room temperature. But mathematically, it is not true. The rate of change of the water temperature depends on the temperature of the water. Thus, we need differential equations to find what would be the temperature after a given time or how much time would it take the water to reach a chosen temperature.
Ways to Solve Them
Let us take an example to solve a differential equation. There is a water filter with a hole at the bottom of the arrangement. Every day 25% of the water goes out of the hole. We have to find the rate of change of height of the water level. So here is the differential equation:
-dH/dt = .25H
Here, dH/dt is the change of the height of water level per day. H is the height of water in the filter. We can also see that water loss each day due to the hole is one fourth of the total water level. The negative sign indicates the loss of water or decrease in value. Thus, the solution of differential equation will be:
H = h*e-.25t
So, we can find the new height of the water ‘H’ after ‘t’ days when the water level in the filter was at a height of h.
What are the Linear Forms?
In usual cases, this kind of equation has a degree one. One side consists of the dy/dx added with a particular constant. It can either be y or x. On the other side, you will see there is a function of that variable only. You have to keep the dy/dx on one side and get the function. Then keep either the ‘dy’ or dx with that function and integrate it. The process is extremely simple as you just have to satisfy the integrating terms. If you see the formula for the temperature difference of water, it involves this type of equation.
Where Can You Apply These?
Whenever you see, there is a growth or change in nature, you can apply this method. It allows you to find out the rate of any function instantly. Even in banks, the officers may use this process to see changes in investments. Differential equations are used in engineering and many other practical applications. Students can learn different derivative formula from the Cuemath website today.