A differential equation is a mathematical equation that relates a function to its derivatives, come into play in a variety of applications, such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behaviour of systems by capturing the rate of change of a quantity and how it depends on other variables.
In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.
A differential equation is an equation involving the derivatives of the dependent variable concerning the independent variable. For example
\frac{d^{2}y}{dx} + x = 0
Here, x is the independent variable, and y is the dependent variable.
A differential equation that includes derivatives concerning only one independent variable is called an ordinary differential equation. There also exist some differential equations which have derivatives for more than one independent variable; they are called partial differential equations.
Example:
Note: Following notations are also used for denoting higher order derivatives.
y' = \frac{dy}{dx} = y_{1} y''' = \frac{d^{3}y}{dx} = y_{3}
Order of a Differential Equation
The order of differential equations is the highest order of the derivative present in the equations.
For example:
x + \frac{dy}{dx} = 3 . It has an order of 1.\frac{d^{2}y}{dx} + sinxcosx = 10 . It has an order of 2.\frac{d^{3}y}{dx} + \frac{d^{2}y}{dx} + x^{3} + 5 = 0 . It has an order of 3.
Degree of Differential Equation
The degree of a differential equation(when it is a polynomial equation in derivatives) is the highest power (positive integral index) of the highest-order derivative involved in the given differential equation.
Examples:
-
(\frac{dy}{dx})^{2} +\frac{d^{2}y}{dx} + 5 = 0 .
Highest order derivative :\frac{d^{2}y}{dx} and the degree of differential equation is 1. \left( \frac{d^2y}{dx^2} \right)^3 + \frac{dy}{dx} = 0 .
Highest order derivative :\frac{d^{2}y}{dx} and the degree of differential equation is 3.\frac{d^3y}{dx^3} + 5\left(\frac{d^2y}{dx^2}\right)^2 - 3\frac{dy}{dx} + y = 0
Highest order derivative is\frac{d^3y}{dx^3} and the degree of differential equation is 1.
Note: It is not always necessary that degree and order of a differential equation are equal, but both of them must be positive.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
General And Particular Solution of Differential Equation
General Solution of Differential Equation : The general solution is a complete form of the solution. It contains one or more arbitrary constants (or functions, in some cases) which can take any value. The general solution represent the family of all possible solutions to the differential equation.
Particular Solution of Differential Equation : A particular solution is obtained when some initial conditions or boundary conditions are provided, that allow us to find the exact value of the arbitrary constants in the general solution.
- Consider a differential equation,
\frac{d^{2}y}{dx^{2}} + y = 0
The solution of this differential equation is a function
- The curve
y = \phi(x) is called the solution of the differential equation. Let's say, this function is,
y = \phi(x) = a cos(x + b)
When this function and its derivatives are substituted in the differential equation, the equation is satisfied and this function is called as general solution of differential equation.
- Let's assume that we gave some values to "a" and "b", where a = 2 and b = -1. Then the equation becomes,
y = \phi_{1}(x) = 2 cos(x -1 )
This is called a particular solution which consisted specific values of arbitrary constants of a and b defined in the General Solution.
Formation of a Differential Equation When General Solution is Given
Let's look at the steps/procedure to form a differential equation from a general solution:
- When the solution contains one parameter
If a family of curves has one constant (say a), it can be written as:
F(x,y,a)=0
Differentiate once with respect to x.
Then eliminate the constant a between the original equation and the differentiated one.
The resulting equation is the required differential equation.
- When the solution contains two parameters
If the family involves two constants (say a and b), write:
F(x,y,a,b)=0
Differentiate twice with respect to x.
Use all three equations to eliminate a and b.
The final equation containing only x, y, and derivatives of y is the required differential equation.
Steps to generate differential equation whose general solution is given
Step 1: Identify the general solution. Start with the general solution of the equation.
y = C_1 e^{2x} + C_2 e^{-2x} Step 2: Differentiate the general solution. Differentiate the general solution with respect to the independent variable (usually x) to eliminate the arbitrary constants. The number of times you differentiate depends on how many arbitrary constants are present. If there are two constants, differentiate twice, and so on.
\frac{dy}{dx} = 2C_1 e^{2x} - 2C_2 e^{-2x} If necessary, differentiate again:
\frac{d^2y}{dx^2} = 4C_1 e^{2x} + 4C_2 e^{-2x} Step 3: Eliminate the arbitrary constants. Use the derivatives to substitute for the arbitrary constants.
\frac{d^2y}{dx^2} = 4y Step 4: Form the differential equation. After eliminating the constants, express the relationship between the function and its derivatives in the form of a differential equation.
\frac{d^2y}{dx^2} - 4y = 0 Step 5: Verify the result. verify the differential equation by checking if the general solution satisfies it.
Question 1: Form the differential equation representing the family of curves y = mx, where, m is an arbitrary constant.
Solution:
We have y = mx,
Differentiating it both sides,
\\ \frac{dy}{dx} = m, Substituting the value of m in the original equation,
y = \frac{dy}{dx}x \\ y - \frac{dy}{dx}x = 0
Question 2: Form the differential equation representing the family of ellipses having foci on the x-axis and centre at the origin.
Solution:
Equation of ellipses with foci on x-axis and centre at origin,
\\ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 Differentiating equations w.r.t x,
\\ \frac{2x}{a^2} + \frac{2ydy}{b^2dx } = 0 \\ \frac{y}{x}(\frac{dy}{dx}) = \frac{-b^{2}}{a^{2}} Differentiating both sides again we get,
\\ xy\frac{d^2y}{dx^2} + x(\frac{dy}{dx})^2 -y\frac{dy}{dx} = 0 This is the required differential equation.
Homogeneous Differential Equations
A homogeneous differential equation is a type of differential equation in which every term is either a multiple of the dependent variable (and its derivatives) or it equals zero.
A function f(x, y) is called a homogeneous function of degree n if,
F(ax, ay) = anF(x, y)
for any constant "a".
A differential equation of the form
Question: Check whether the differential equation,
Solution:
\frac{dy}{dx} = \frac{x + 2y}{x - y} Let,
\\ F(x,y) = \frac{x + 2y}{x-y} Let a be a constant,
\\ F(ax,ay) = \frac{ax + 2ay}{ax - ay}\\ = \frac{x + 2y}{x - y}a^{0}\\ = a^{0}F(x,y) Since this function is homogeneous, the differential equation is also homogeneous.
Variable Separable Differential Equation
A variable separable differential equation (or separable differential equation) is a type of first-order differential equation that can be rewritten in such a way that all terms involving the dependent variable y are on one side of the equation and all terms involving the independent variable x are on the other side.
Consider a first-order differential equation of the form,
If F(x, y) can be expressed as h(x)g(y) where h(x) is a function of x and g(x) is a function of y. Then the equation is called a differential equation of variable separable type. The differential equation has the form,
Question: Find the general solution of the differential equation,
Solution:
\frac{dy}{dx} = \frac{x+1}{2-y} \\ (2 - y)dy = (x +1)dx \\ \int (2 - y)dy = \int (x + 1)dx \\ 2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C \\ 4y - y^2 -x^2 -4x = C
Solution to a Linear Differential Equation
A linear differential equation is a differential equation that can be made to look like in this form:
where P(x) and Q(x) are the functions of x. It is solved using a special approach:
- Make two new functions of x, call them u and v, and say that y = uv.
- Then solve to find u, and then v.
Step-by-step procedure:
Step 1: Substitute y = uv, and
\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} into,
\frac{dy}{dx} + P(x)y = Q(x) Step 2: Now, one should factor the parts that have "v".
Step 3: Put the v term equal to zero (this gives a differential equation in u and x which can be solved in the next step) Put the v term equal to zero (this gives a differential equation in u and x which can be solved in the next step)
Step 4: Solve "u" and then put it back in the equation to find "v".
Step 5: Finally, y = uv is the solution.
Let's look at an example to understand it better,
Question: Solve
Solution:
This is a linear equation. Let's bring it in the form specified above.
\frac{dy}{dx} + P(x)y = Q(x) Here, P(x) = -1/x and Q(x) = 1.
So, let's follow the steps given above. Substitute y = uv in the equation. Then the equation becomes,
u\frac{dv}{dx} + v\frac{du}{dx} - \frac{uv}{x} = 1 \\ = u\frac{dv}{dx} + v(\frac{du}{dx} - \frac{u}{x}) = 1 Put the parts involving "v" equal to zero.
\frac{du}{dx} - \frac{u}{x} = 0\\ = \frac{du}{u} = \frac{dx}{x} \\ = \int \frac{du}{u} = \int \frac{dx}{x} \\ = ln(u) = ln(x) + C \\ = ln(u) = ln(x) + ln(k) \\ = ln(u) = ln(xk) \\ = u = xk Substituting "u" back into the equation.
kx\frac{dv}{dx} = 1 Now, let's solve this to find "v".
kx\frac{dv}{dx} = 1 \\ = dv = \frac{1}{k}\frac{dx}{x}\\ = \int dv = \int \frac{1}{k}\frac{dx}{x} \\ = v = \frac{ln(x)}{k} + ln(c) \\ = v= \frac{ln(xc)}{k} Substitute both "u" and "v" into the equations y = uv.
y = uv \\ = kx \frac{1}{k}ln(cx)\\ = xln(cx) So, this is the solution for this differential equation.
Writing a Differential Equation
Now let's move on to modelling a differential equation. Modelling is the process of writing a differential equation that describes a physical situation. We will see how to model first-order differential equations, modelling more complex orders is out of scope at this level of study.
Example: Savings Account Model
Write x(t) for the number of dollars in the account at time t. It accrues interest at an interest rate r. The interest rate has units of percent/year. The more money in the account the more interest you earn. At the end of an interest period of Δt years (e.g. Δt = 1/12, or Δt = 1/365) the bank adds "r.x(t)·Δt" dollars to your account. This means the change Δx in your account is
Δx = r.x(t).Δt r has units of (years)−1.
Mathematicians and some bankers like to take things to the limit. Rewrite our equation as
\frac{Δx}{\Delta t} = rx(t) , and suppose that the interest period is made to get smaller and smaller. In the limit as Δt → 0, we get the differential equation
\dot{x} = rx Now suppose we make contributions to this savings account. We’ll record this by giving the rate of savings, q. This rate has units dollars per year, so if you contribute every month then the monthly payments will be q Δt with Δt = 1/12. This payment also adds to your account, so, when we divide by Δt and take the limit, we get
\dot{x} = rx+q. This is a linear differential equation.
Practice Question on Differential Equation
Question 1 : Solve the differential equation:
Question 2: Find the differential equation for the given general solution
Question 3: Solve the second-order differential equation:
Question 4: Solve the homogeneous differential equation: