Differential Equations — Part 1: Order, degree, solutions and separable equations: visual lesson, CBSE Class 12 Mathematics

Lesson

Differential Equations, Part 1

Equations whose unknown is a function.

In this lesson you will

Find the order and degree of a differential equation, and recognise when the degree is not defined
Distinguish general and particular solutions and verify a solution
Solve equations with separable variables
Use separable equations for growth and decay

$C = -2$
$C = 0$
$C = 2$

Solutions of $\dfrac{dy}{dx} = 2x$: the general solution is a family; the condition $y(1) = 3$ picks $C = 2$.

Concept

What you will be able to do

Find the order and degree of a differential equation
Tell general and particular solutions apart and verify a solution
Solve equations by separating the variables
Model growth and decay

Rates of change, written as equations.
Situation	Equation
Bacteria in a lab culture	$\dfrac{dN}{dt} = kN$
A cup of tea cooling	$\dfrac{dT}{dt} = -k(T - T_0)$
A drug entering the blood by a drip	$\dfrac{dC}{dt} + 0.5C = 2$
A curve with a given slope rule	$\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{2xy}$

Concept

What is a differential equation?

A differential equation is an equation involving an unknown function and one or more of its derivatives, such as $\dfrac{dy}{dx} = 2x$ or $\dfrac{d^2y}{dx^2} + 4y = 0$ .

Here $y$ depends on one variable $x$ : an ordinary differential equation
We write $y'$ , $y''$ for $\dfrac{dy}{dx}$ , $\dfrac{d^2y}{dx^2}$

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