Ordinary Differential Equations: Overview, Questions, Preparation

In Mathematics, an ordinary differential equation (also abbreviated as ODE) is an equation that consists of one or more functions along with their derivatives of one independent variable. An equation that includes a function with one or more derivatives is a differential equation. In the case of ODE, however, the term ordinary is used for the derivative of a single independent variable function.
In some types of differential equations, more than one vector could have derivatives for functions. The forms of DEs are partial differential equations, homogeneous and non-homogeneous differential equations, linear and nonlinear differential equations.

ODE Differentiation

In mathematics, the word 'Ordinary Differential Equations' also known as ODE is an equation containing only one independent variable with respect to the variable and one or more of its derivatives. In other words, the relationship with one independent variable x, the real dependent variable y, with some of its derivatives, is represented as the ODE.
y',y’', ....y’n,...with regard to x.

Application

ODEs have excellent software that has the potential to foresee the world around them. It is used in a number of areas, such as medicine, economics, chemistry, mechanics, and engineering. This helps to forecast the rise and decline of exponential growth, population, and species growth. Any of ODEs' uses are:

1. The modelling of epidemic progression
2. Describes how energy travels
3. Describes the pendulum movement, waves
4. Used in Newton's Second Motion Law and the Cooling Law

In Class 12, in the chapter differential equation, you will get to learn about ode in detail, along with other properties. The weightage of this chapter is 5-6 marks.

1. Solve y’=4x+1
Solution.

Given, y’=4x+1

Integrate on both sides,

∫ y’dx = ∫ (4x+1)dx

y = 4x³/4 + x + C

y =x³ + x + C

Where C is an arbitrary constant.

2. Prove y=√1+x2 : y’=xy/1+x²
Solution.

y’= d/dx (√1+x²)

y’= 1/2√1+x²). d/dx (1+x²))

y’= 2x/2√1+x²

y’= x/√1+x²

y’= x/1+x² x √1+x²

y’= x/1+x². y

y’= xy/1+x²

Therefore, LHS=RHS.

3. Find the differential equation for x/a+y/b=1
Solution.

On differentiating w.r.t  x

1/a+1/b dy/dx=0

1/a+1/b y’=0

Differentiating again w.r.t x

0+1/b y”=0

Ans: y”=0

Q: What do ordinary differential equations mean?

Q: What are PDE and ODE?

Q: Are ordinary differential equations difficult?

Q: What does satisfying a differential equation mean?

Q: Where are ordinary equations used for differentials?