Class 12 - Mathematics - Differential Equations - Study Notes - Part - 1 #calculus #differentialEquations #class12Maths #eduvictors

Class 12 - Mathematics - Differential Equations - Part - 1

An equation involving an independent variable (variables), dependent variable and derivative or derivatives of the dependent variable with respect to the independent variable (variables) is called a differential equation.

It is an equation involving unknown functions and their derivatives w.r.t. one or more independent variables.

e.g. $ 2x \frac{\mathrm{d} y}{\mathrm{d} x} - 3y = 6$  and $ 4\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}} + \left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{3} $ = 0 are differential equations.

Q1: Is  5x + 7y = 0 a differential equation?

Answer: No, it is not a differential equation because derivatives of the dependent variable (y) with respect to the independent variable (x) are not present.

Order of a Differential Equation

The order of the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation is called the order of the differential equation.

The order of a differential equation is the same as that of the highest derivative (or differential) it contains.

e.g. 

(i) $ \frac{\mathrm{d} y}{\mathrm{d} x} - 3\sin x = 0 $ 

(ii) $ \frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}} + 2x^{3}\left ( \frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}} \right ) = 0 $

Here, said above  (i), the equation has the highest derivative of the first order and in e.g. (ii), the equation has the highest derivative of the third order. 

So, the orders of the differential equations in e.g. (i) and (ii) are 1 and 3, respectively.

Degree of a Differential Equation

The degree of a differential equation whose terms are polynomials in the derivatives is defined as the highest power (positive integral index) of the highest-order derivative in it after the equation is freed from radicals and fractions in its derivatives.

The highest power (positive integral index) of the highest order derivative involved in a differential equation, when it is written as a polynomial in derivatives, is called the degree of a differential equation.

e.g. $ \left (\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}  \right )^{2} + x\left (\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}  \right ) + 3y\left (\frac{\mathrm{d} y}{\mathrm{d} x}  \right ) = 0 $

In this differential equation, highest order derivative is $ \left (\frac{\mathrm{d}^{3} y}{\mathrm{d} x^{3}}  \right ) $, whose highest power is 2. So, degree of differential equation is 2.

  

Linear and Non-Linear Differential Equations

A differential equation is said to be linear if the unknown function and its derivative, which occur in the equation, occur only in the first degree and are not multiplied together. Otherwise, the differential equation is said to be non-linear.

e.g. $\frac{\mathrm{d}y }{\mathrm{d} x} = \sin x$ is a linear,

$(\frac{\mathrm{d^2}y }{\mathrm{d} x^2})^2 + x^2 (\frac{\mathrm{d} y}{\mathrm{d} x})^3 = 0$ is non-linear one.

Note: A linear differential equation is always of first degree but the converse is not true i.e. every differential equation of the first degree may not be linear.

Solution of Different Forms of Differential Equations 

(i) If the equation is:

$\frac{\mathrm{d} y}{\mathrm{d} x} = f(x)$ then y = $\int f(x)dx + c $

(ii) Variables separable. If the equation is

$\frac{\mathrm{d}y }{\mathrm{d} x} =f(x)g(y)$, then $\int \frac{\mathrm{d}y }{g(y)} = \int f(x) + c$

(iii) Reducible to variables separable.

If the equation is $\frac{\mathrm{d} y}{\mathrm{d} x} = f(ax + by + c)$ then put ax + by + c = z.

(iv) Homogenous equation

A function f(x, y) is said to be homogenous of degree n if f(λx, λy) = λⁿf(x,y) for any non-zero constant λ.

If the equation is $\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{f(x,y)}{g(x,y)}$, where f(x,y), g(x,y) are homogenous functions of same degree in x and y. Put y = vx.

(v) Linear Equation

If the equation is $\frac{\mathrm{d} y}{\mathrm{d} x} + Px = Q$, where P, Q are constants or functions of x, then $ye^{\int Pdx} = \int Qe^{\int Pdx} dx + c$, where $\int Pdx$ is the integrating factor.

In the next post, we'll solve problems related to differential equations.

👉See Also:

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