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Ordinary Differential Equations

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Ordinary Differential Equations

An ordinary differential equation (ODE) is an equation that describes the relationship between a function and its derivatives. In other words, it relates a function to its rate of change.

Differential equations of first-order and first degree

Definition: Differential equations are equations with variables and their differentials.

There are two different kinds of them

1. Ordinary differential equations

2. Partial differential equations

Ordinary differential equations

If one or more variables can be differentiated with respect to just one independent variable, an equation is said to be ordinary.

Example: $\frac{dy}{dx} + 9 xy = x^{2}$ , $\frac{d^{2} y}{{dx}^{2}} + 5 \frac{dy}{dx} + 8 y = e^{x}$

Try Yourself 1!

Which of the following are ordinary differential equations?

		$\frac{dy}{dx} + 6 xy = x^{2}$
		$3 \frac{dy}{dx} + \frac{d^{2} x}{{dy}^{2}} = x^{2}$
		$\frac{dy}{dx} + \frac{d^{2} y}{{dx}^{2}} = 4 x^{2}$
		$\frac{dy}{dx} + \frac{dx}{dy} = 5 x^{2}$

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Partial Grading Explained

Partial Differential Equation

If the derivatives of a differential equation refer to two or more independent variables, the equation is said to be partial.

Example: ${(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} = 5 z$ , $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 6 z$

Try Yourself 2!

Which of the following are Partial differential equation?

		$\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 6 z + xy$
		$\frac{dy}{dx} = \frac{d^{2} y}{{dx}^{2}} - 7 x^{2}$
		$\frac{dy}{dx} + \frac{dx}{dy} = x^{2}$
		$y \frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = 0$

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Partial Grading Explained

Order of a differential equation

In the differential equation, it is the order of the highest derivative that occurs. If the n^th derivative is the equation's highest derivative, the differential equation is said to be of order "n".

Example:

1. $(x^{3} + 2) \frac{dy}{dx} + 5 xy = 6 x^{2}$

The order of this Differential equation is $1$ .

2. $x \frac{d^{2} y}{{dx}^{2}} - (3 x + 4) \frac{dy}{dx} + (x + 2) y = e^{x}$

The order of this Differential equation is $2$ .

Try Yourself 3!

Find the order of the following differential equations.

$4 (\frac{d^{6} y}{d x^{6}}) - {(\frac{d^{3} y}{{dx}^{3}})}^{3} + \frac{5 dy}{dx} = 4$

Order=

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Degree of a differential equation

The degree of a differential Equation is the highest degree of the highest derivative in the equation, after the equation is made free from radicals and fractions in its derivations.

Example:

1. $y = x \frac{dy}{dx} + \sqrt{1 + {(\frac{dy}{dx})}^{2}}$ on solving we get $(1 - x^{2}) {(\frac{dy}{dx})}^{2} 2 xy \frac{dy}{dx} + (1 - y^{2}) = 0$

Degree $= 2$ .

2. $a \frac{d^{2} y}{{dx}^{2}} = {[1 + {(\frac{dy}{dx})}^{2}]}^{\frac{3}{2}}$ on solving we get $a^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2} = {[1 + {(\frac{dy}{dx})}^{2}]}^{3}$

Degree $= 2$ .

Try Yourself 4!

Find the degree of the following differential equations.

$4 {(\frac{d^{3} y}{d x^{3}})}^{3} - {(\frac{d^{2} y}{{dx}^{2}})}^{2} + \frac{5 dy}{dx} = 4$

Degree=

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