First-Order Differential Equation Solution
First-Order Differential Equation is generally solved and simplified using two methods mentioned below.
- Using Integrating Factor
- Method of variation of constant
Integrating Factor Homogenous Differential Equation
The integrating factor is a function used to solve first-order differential equations. It is most commonly applied to ordinary linear differential equations of the first order. If a linear differential equation is written in the standard form y’ + a(x)y = 0
Then, the integrating factor (μ) is defined as: [Tex](\mu = e^{\int P(x)dx})[/Tex]
Solution using Integration Factor
Using integrating factor can be used to simplify and facilitate the solution of linear differential equations. The entire equation becomes exact when the integrating factor, which is a function of x, is multiplied.
For any linear differential equation is written in the standard form as:
y’ + a(x)y = 0
Then, the integrating factor is defined as:
u(x) = e(∫a(x)dx)
Multiplication of integrating factor u(x) to the left side of the equation converts the left side into the derivative of the product y(x).u(x). General solution of the differential equation is:
y = {∫u(x).f(x)dx + c}/u(x)
where C is an arbitrary constant.
Method of Variation of a Constant
Method of Variation of a Constant is a similar method to solve first order differential equation. In first step, we need to do y’ + a(x)y = 0. In this method of solving first order differential equation, homogeneous equation always contains a constant of integration C.
- In solving non-homogeneous linear differential equations, we replace the constant (C) with an unknown function (C(x)).
- By substituting this solution into the non-homogeneous equation, we can determine the specific form of the function (C(x)).
- This approach allows us to find a particular solution that satisfies the given non-homogeneous equation.
- Interestingly, both the method of variation of constants and the method of integrating factors lead to the same solution.
Remember, this technique helps us handle non-homogeneous differential equations by introducing a function that varies with the independent variable!
First Order Differential Equation
A first-order differential equation is a type of differential equation that involves derivatives of the first degree (first derivatives) of a function. It does not involve higher derivatives. It can generally be expressed in the form: dy/dx = f(x, y). Here, y is a function of x, and f(x, y) is a function that involves x and y.
It defined by an equation dy/dx = f (x, y) where x and y are two variables and f(x, y) are two functions. It is defined as a region in the xy plane. These types of equations have only the first derivative dy/dx so that the equation is of the first order and no higher-order derivatives exist.
Differential equations of first order is written as;
y’ = f (x, y)
(d/dx)y = f (x, y)
Let’s learn more about First-order Differential Equations, types, and examples of First-order Differential equations in detail below.
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