Ordinary Differential Equations

Order of ODEs

The order of an ODE is the order of the highest order derivative present in the equation.

The general form for an n-th order ODE can be represented as a function of x, y, y', y"… yⁿ ,i.e., F(x, y, y', y"… yⁿ) = 0.

Here yⁿ is the highest order derivative and decides the order of the ODE.

There are 3 subtypes of ODEs.

1. Linear Ordinary Differential Equations.
2. Autonomous Ordinary Differential Equations.
3. Non-linear Ordinary Differential Equations.

Linear ordinary differential equations: 

These can be written in the form p_n(x)yⁿ + p_n-1(x)y^n-1 + … p₁(x)y + p₀(x) = 0. Here p_i(x) for 0 <= x <= n are referred to as the coefficients of the linear differential equation. These equations often occur naturally (in simple pendulum, Law of cooling, etc.) and have various applications.

The linear differential equations can be further subdivided into two more categories.

1. Homogeneous differential equations.
2. Non-homogeneous differential equations.

In the equation for a linear differential equation discussed earlier, i.e., p_n(x)yⁿ + p_n-1(x)y^n-1 + … p₁(x)y + p₀(x) = 0. If p₀(x) = 0 then the equation is referred to as a homogeneous differential equation and if p₀(x) ≠ 0 then it’s referred to as a non-homogeneous differential equation.

Autonomous ordinary differential equations:

It is an ordinary differential equation that does not explicitly depend on the independent variable.

Non-linear ordinary differential equations:

If a differential equation cannot be written as a linear combination of derivatives of y, i.e., F(x, y, y', y"...) can't describe a non-linear differential equation.

Examples

Example. 1: solve y’ = y + x, y(0) = 0.

=> multiply both sides by e^-x.

=> y’e^-x - ye^-x = xe^-x

=> (ye^-x)’ = xe^-x

=> ye-x = ∫xe^-x + C₁

=> ye^-x = -xe^-x - e^-x + C₂

We obtained the above solution, substitute y(0) = 0.

0 = 0 - 1 + C₂

C₂ = 1

Ans.

=> ye^-x = -xe^-x - e^-x + 1

=> y = -x - 1 + e^x (as e^-x != 0 for all x ∊ R)

 

Example. 2: solve y’ = x + 1, y(0) = 1.

=> dy/dx = x + 1

=> dy = (x + 1)dx

=> ∫dy = ∫(x+1)dx

=> y + C₁ = x²/2 + x + C₂

=> y = x²/2 + x + C₃

We obtained the above solution, substitute y(0) = 1.

1 = 0 + 0 + C₃

C₃ = 1

Ans.

=> y = x²/2 + x + 1

FAQs

1. What are some examples of ODEs?

 Following are some examples of ODEs.

1. y’ = 0.

2. y’’ + xy’ = 0

3. y’’ + xy’ + x = 0

4. y’’ - ky’ - sin(x) = 0

5. y’’’ + y’’ + y’ + y = 1729

2. What are the different types of ODEs?

ODEs are divided into 3 sub-types.

a) Linear Ordinary Differential Equations.

b) Autonomous Ordinary Differential Equations.

c) Non-linear Ordinary Differential Equations.

3. What is the order of an ODE?

The order of an ODE is the order of the highest order derivative present in the equation.

4. Why should someone learn about ODEs?

Questions related to ODEs are often asked in various competitive exams such as GATE. This topic is also foundational before one discovers more complicated topics such as Navier Stokes.

5. Describe some common applications of ODEs?

ODEs are often used to model and describe simple pendulums (earlier clocks). They are also used extensively to model the behaviour of various dynamic systems, such as the growing bacterial population.

6. Difference between ODE (Ordinary Differential Equations) and PDE (Partial Differential Equations)?

ODEs refer to equations where the derivatives are taken, w.r.t. only one variable. That is, there is only one independent variable. PDEs refer to equations that depend on partial derivatives of several variables.

Key Takeaways

In this article, we have extensively discussed Ordinary differential equations, their types, some examples, and their applications. After going through the article, refer to the following articles on linear differential equations and non-homogeneous differential equations. Also, refer to articles on Partial Differential Equations and System of Linear Equations after referring to the earlier articles.

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