The importance of integration
Watch this to unpack how to solve a differential equation with integration in Higher Maths (and to appreciate why there are so many of us)
The world's human population doesn’t stay the same. In fact, over the last few hundred years it has been growing faster and faster. During the 1st century, it was estimated to be between 150 and 300 million. And now in the 21st century it’s over 7 billion.
Although the rate of growth is slowing slightly, there are so many young people that the population will continue to grow for some time. Using mathematical modelling as a guide it is believed that the population might increase to 9.9 billion by 2050.
It doesn’t matter whether you are looking at human populations or of other species, the most useful way to describe a population is with a rate of change and so the equation ends up being a differential one, one that contains a derivative.
Many fundamental laws of physics and chemistry can be formulated as differential equations, and in biology and economics, they are used to model the behavior of many complex systems. So, this stuff really matters.
We’re going to look at how to solve a simple differential equation using integration.
This is thinkfour.
This is a differential equation. It contains a derivative, the first derivative. As the derivative dy by dx is a function of x we can integrate it to find y.
dy by dx is 6x squared -3x + 4.
To find y we need to integrate, so we write that y equals the integral of six x squared -3x + 4 dx. We need the y equals, we need the integral sign and we need the dx.
Then we integrate and y = 2x cubed -3/2 x squared + 4x + C.
The plus C is essential here, as we are going to find it as the next step.
Next we will use information given to us to find the value of C.
Suppose we are told that y = 14 when x = 2. We can substitute these values in for x and y and solve the resulting equation to find the c value.
If we do this c = -4.
So we have our function y = 2x cubed - 3/2 x squared + 4x - 4.
The use of notation and layout of method is really important here.
Here is another question involving the same technique.
The rate of change of the volume of an object with respect to time in hours is given by dV by dt equals 9t squared - 10t.
After 2 hours the Volume of the object is 30.
We need to find the formula for the Volume of the object.
We can find volume by integrating dV by dt.
We write that V equals the integral of 9t squared - 10t, dt.
Then we integrate and V equals 3t cubed -5t squared plus C.
Now we can use the information given to find C, by substituting t = 2 and V = 30 into the derivative.
This gives C as 26.
So the volume is equal to 3t cubed -5t squared +26.
Being able to solve differential equations is one of the reasons that integration is so important to us.
By understanding differential equations we can start to understand population growth, and decline, and so be better placed to address some of the biggest issues on the planet right now.
So, don’t give up, it is worth getting right. You never know, you might just need this one day for some very important decision-making.
This was thinkfour, thanks for watching.