Previously we learnt about . In this chapter we will learn about a very important application of integration, which is solving differential equations.
I. Introduction
Let us try to understand what is a differential equation using an example.
Consider the equation $y^2 = x\cdot \sin y$
Differentiating both sides w.r.t. $x$, we get:
$\dfrac{dy^2}{dx} = \dfrac{d(x\cdot \sin y)}{dx}$
$\Rightarrow \dfrac{dy^2}{dy} \cdot \dfrac{dy}{dx} = \sin y + x \dfrac{d (\sin y)}{dx}$
$\Rightarrow 2y \cdot \dfrac{dy}{dx} = \sin y + x \dfrac{d (\sin y)}{dy} \cdot \dfrac{dy}{dx}$
$\Rightarrow 2y \cdot \dfrac{dy}{dx} = \sin y + x \cos y \cdot \dfrac{dy}{dx}$
$\Rightarrow 2y \cdot \dfrac{dy}{dx} - x \cos y \cdot \dfrac{dy}{dx} = \sin y$
$\Rightarrow (2y - x \cos y) \dfrac{dy}{dx} = \sin y$
$\Rightarrow \dfrac{dy}{dx} = \dfrac{\sin y}{2y - x \cos y}$
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The above is an example of differential equation. When we solve this equation we should get a family of equations, represented by:
$y^2 = x\cdot \sin y + c$, where $c$ is a constant.
Differentiating any equation of this family will give us the same differential equation. To find the exact equation of this family we will need some initial value. For example, if we are given the fact that for $x = 0$, $y = 0$ we should get the value of $c$ as $0$.
II. Applications Of Differential Equation
Before we get onto understanding further details about how to solve differential equations, we need to understand why do we need differential equations at all. If we already know the function which led to the differential equation as shown in the previous section, then we do not need a method to solve for differential equation.
However, a significant number real life problems start by defining the rate of change of a variable w.r.t some independent variable, and subsequently we solve this to get the actual relationship.
Let us take a couple of examples:
Newton's second law states that the rate of change of velocity is proportional to the applied force. Therefore, our definition leads to the equation:
$\dfrac{dv}{dt} \propto F$. Also, we know that the force required is proportional to the mass of the body, $F \propto m$. Combining the two, we get:
$F = k \cdot m \cdot \dfrac{dv}{dt}$. Of course, we defined the unit $Newton$ such that $k$ became $1$, and thus we got $F = m\dfrac{dv}{dt}$. All other equations of motion, where force may or may not be constant, can be obtained by solving the above differential equation.
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Similarly, if we are given the fact that the rate of growth of population of a given organism is proportional to the total population of that organism in a given environment. This leads to the equation:
$\dfrac{dy}{dt} \propto y \Rightarrow \dfrac{dy}{dt} = ky$, where $y$ represents the total population of the organism.
Now, if we were to estimate the total population of the organism at any given time $T$, we can do so only by solving the above differential equation.
Because of this, differential equations is considered an important tool in solving many real life problems in various fields like physics, chemistry, biology, etc.
III. Types Of Differential Equation
Differential equations can be broadly classified into two types, viz:
Ordinary Differential Equation
If the dependent variable or variables are dependent only on one independent variable, then the equation is said to be ordinary. For example,
$\dfrac{dy}{dx} + \dfrac{dz}{dx} = y + z$, where $y = f(x)$ and $z = g(x)$, is an ordinary differential equation.
Partial Differential Equation
If the dependent variable or variables are dependent on two or more independent variables, the the equation is said to be partial differential equation. For example:
It is the highest order derivative appearing in the equation.
Degree
It is the highest power of the highest order derivative appearing in the equation expressed in standard form, without any radicals of the derivative terms.
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V. Family Of Curves
One major use of differential equations is to represent a family of curves without using any arbitrary constant.
Let us understand this using an example. Consider the equations of all parabolas passing through the origin and having their focii on the $y$-axis.
This is given by $y = ax^2$
No matter what value of $a$ we choose, the curve will always pass through the point $(0, 0)$ and for every value of $y$ there will be two values of $x$ given by $\pm \sqrt{\dfrac{x}{a}}$. This means the curve will be symmetric about the $y$-axis, therefore, the focus will lie on the $y$-axis.
If we were to represent the same family of curves without using the arbitrary constant $a$ then what is the way?
We can use differential equations to eliminate the arbitrary constant, and get a representation of the family of curves purely in terms of $x$ and $y$.
Differentiating the equation $y = ax^2$ once, we get:
$\dfrac{dy}{dx} = 2ax$
$\Rightarrow a = \dfrac{1}{2x}\dfrac{dy}{dx}$
Substituting the above value of $a$ in the original equation, we get:
$y = \dfrac{1}{2x}\dfrac{dy}{dx} x^2$
$\Rightarrow y = \dfrac{1}{2} \dfrac{dy}{dx} x$
$\Rightarrow x\dfrac{dy}{dx} - 2y = 0$
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$\Rightarrow \dfrac{dy}{dx} = \dfrac{2y}{x}$
This tells us that for any curve belonging to this family, the slope at any point $(x, y)$ of the curve, is twice the ratio of the $y$ and $x$ values at that point.
Let us take another example.
If we consider all rectangular hyperbolas whose asymptotes are the two axes, that is $xy = k$, then by differentiating w.r.t. $x$ we get:
$y + x\dfrac{dy}{dx} = 0$
$\Rightarrow \dfrac{dy}{dx} = - \dfrac{y}{x}$
So, we can draw the conclusion that for any curve belonging to this family, at any point $(x, y)$ the slope at that point will be the negative of the ratios of the $y$ and $x$ values.
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