A first-order differential equation is defined by an equation
Â dy/dx =f (x,y)
of two variables x and y with its function f(x,y) defined on a region in the xy-plane. It has only the first derivative dy/dx,Â so that the equation is of first order and not higher-order derivatives.
The above first order differential equation can also be written as
yâ€™ = f (x,y) Â Â or
(d/dx) y = f (x,y)
The differential equation is generally used to express a relation between the function and its derivatives. In Physics and chemistry, it is used as a technique for determining the functions Â over its domain if we know the functions and some of the derivatives.
First Order Linear Differential Equation
If the function f is a linear expression in y,then the first order differential equation yâ€™ = f (x,y) is a linear equation. That is, the equation is linear and the function f takes the form
f(x,y) = p(x)y + q(x)
Since the linear function is y = mx+b
where p and q are continuous functions on some interval I. Differential equations that are not linear are called nonlinear equations.
Consider the first order differential equation yâ€™ = f (x,y) is a linear equation and it can be written in the form
yâ€™ + a(x)y = f(x)
where a(x) and f(x) are continuous functions of x
The alternate method to represent the first order linear equation in a reduced form is
(dy/dx) + P(x)y = Q (x)
Where P(x) and Q(x) are the functions of x which are the continuous functions. If P(x) or Q(x) is equal to zero, the differential equation is reduced to the variable separable form. It is easy to solve when the differential equations are in variable separable form.
Method 1 : Using an Integrating Factor
If a linear differential equation is written in the standard form:
yâ€™ + a(x)y = 0
Then, the integrating factor is defined by the formula
\(u(x)=exp(\int a(x)dx)\)Multiplying the integrating factor u(x) on the left side of the equation that converts the left side into the derivative of the product y(x)u(x).
The general solution of the differential equation is expressed as follows:
\(y=\frac{\int u(x)f(x)dx+C}{u(x)}\)where C is an arbitrary constant.
Method 2 : Method of Variation of a Constant
This method is similar to the integrating factor method. Finding the general solution of the homogeneous equation is the first necessary step.
yâ€™ + a(x)y = 0
The general solution of the homogeneous equation always contains a constant of integration C. We can replace the constant C with a certain unknown function C(x). When substituting this solution into the non-homogeneous differential equation, we can be able to determine the function C(x). This approach of algorithm is called the method of variation of a constant. Both the methods lead to the same solution.
Properties of First-order Differential Equations
The Linear first order differential equation possesses the following properties
- It does not have any transcendental functions like trigonometric functions and logarithmic functions
- The products of y and any of its derivatives are not present
Applications of First-order Differential Equation
Some of the applications which use the first-order differential equation are as follows:
- Newtonâ€™s law of cooling
- Growth and decay
- Orthogonal trajectories
- Electrical circuits
- Falling Body Problems
- Dilution Problems
Sample Example
Question :
Solve the equation \({y}’-y-xe^{x}=0\)
Solution :
Given, \({y}’-y-xe^{x}=0\)
Rewrite the given equation and the equation becomes,
\({y}’-y=xe^{x}\)Using the integrating factor, it becomes
\(u(x)=e^{\int (-1)dx}=e^{-\int dx}=e^{-x}\)Therefore, the general solution of the linear equation is
\(y(x)=\frac{\int u(x)f(x)dx+C}{u(x)}=\frac{\int e^{-x}xe^{x}dx+C}{e^{-x}}\) \(y(x)=\frac{\int xdx+C}{e^{-x}}=e^{x}\left ( \frac{x^{2}}{2}+C \right )\).Register with BYJUâ€™S learning app to get more information about the maths related articles and start practice with the problems.
Related Links | |
Homogeneous Differential Equation | Differential Equation and its Types |
Formation of Differential Equations | Methods of Integration |