Differential Equations Class 12 notes | IIT-JEE study material

So, in calculus we are starting the topic of differential equations, we will be seeing what does a differential equation exactly means, what are partial differential equations, the significance of differential equations in various disciplines is very vast, we will look at the very basic understanding of the interpretation of solutions of curves from given differential equation, and how differential equations represent a family of the curve.

Introducing the Differential equations

any equation having terms containing derivative[differentiation] of one, two, or many dependent variables with respect to the independent variable is defined to be a differentiable equation, It should be noted that the discussion we generally made is by assuming y as a function of x that is y is dependent variable and x is an independent variable.

Differential Equations Class 12 notes | IIT-JEE study material
A family of orthogonal curves – credits

further generally we can state equation containing differential coefficients [Latexpage] {$ dx, dy $ are called differential coefficients} of dependent variable with respect to differential coefficients of independent variables is called a differential equation. for example , followings are some differential equations-

$$ \frac{dy}{dx}+\frac{y}{x}=0 $$ $$\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+x=0 $$

Order of differential equation

order of any differential equation is basically the highest order derivative present in the given equation,

$\frac{dy}{dx}$ is called the first order derivative, similarly $\frac{d^{2}y}{dx^{2}}$ is called as second order derivative,

hence differential/derivative of the following form is defined to be a derivative of order “m”

$$ \frac{d^{m}y}{dx^{m}} $$

Degree

If we express the differential equation in polynomial form(non-negative exponents) the degree of the differential equation is defined as the exponent of the highest order derivative present,

$$ {\left(\frac{d^{2}y}{dx^{2}}\right)}^{1}+{\left(\frac{dy}{dx}\right)}^{3}+x+y=0$$

the degree of the above differential equation by definition is an exponent of the highest order derivative hence, the degree is 1,

it is not necessary that every time we can express the differential equation in polynomial form, hence degree can be defined and it can be undefined in some polynomials as well, hence degree need not be defined in every differential equation.

Formation of differential equations

basically, a differential equation represents a family of curves, moreover, the scratch idea is if we differentiate an equation that represents a family of curves we will get the differential equation of the family of the curve(although we will describe the whole process), then if we integrate a given differential equation we will get the equation of the family of curves.

Let’s start with the equation $y=mx$ it represents all the curves which pass through origin(0,0) with different different slope(m), hence here “m” is an arbitrary constant (that is it can take various values for different curves, but for a particular line “m” is constant),

Hence the equation

$$y=mx$$ or say

\begin{align} m=\frac{y}{x}\end{align} consist of one arbitrary constant which is “m”, Now if we differentiate this equation once we get $$dy=m.dx$$ or say

$$\frac{dy}{dx}=m$$

using equation (1)

\begin{align} \frac{dy}{dx}=\frac{y}{x} \end{align}

Equation (2) represents differential equation of family of lines passing through origin which are represented by equation (1).

So, What we had done above is how we can represent any family of curves with a differential equation, so, we can summarize the above process in some steps, so to represent any family of curves via differential equation.

Differential equation formation procedure

it is very necessary to digest the above discussion, as we are writing steps from the procedure we just did in the above discussion,

For any equation representing a family of curves having “m” number of essential * arbitrary constants,

  • differentiate the given equation “m” times, after each differentiation write the obtained equation separately, hence obtain “m” equations.
  • with the help of equation given initially and “m” equations during differentiation, try to eliminate the arbitrary constants by using substitution, and basic mathematics
  • the final equation obtained after eliminating all the arbitrary constants is the required differential equation for the family of curve.

*Essential arbitrary constant means the lowest number of arbitrary constants which can be made possible by grouping the arbitrary constants if possible, For example, one can write a new arbitrary constant $c_{3}$ in place of $c_{1}+c_{2}$ that is

$$c_{3}=c_{1}+c_{2}$$

One should self observe and appreciate that to eliminate “m” essential constants from a equation one need to have atleast total of “m+1” equations.

Hence if we are having a equation having “m” arbitrary constants we differentiate it “m” times to obtain “m” equations and if we include the one equation which we are given initially we have total of “m+1” equations.

Reader Should try on eliminating constants from an equation , and reach to conclusion of “m+1” equations are required.

Solution of differential equations

The solution of a differential equation is the algebraic equation representing the family of curves, basically, the solution of a differential equation is obtained by integrating the differential equation as it’s the reverse process of formation of the differential equation.

Variable separable method

we have various differential equations, in some of them the variables can be separated, these differential equations can be solved by separating the variable with its differential coefficient(dx or dy) and can be integrated to get the algebraic equations of the family of curves.

Homogenous differential equation

any function is said to be a homogenous function of order “n” , in which if x is replaced by $\lambda x$ and y is replaced by $\lambda y$ then if the function satisfies the following :

$$ f(x,y)= \lambda^{n}f(x,y) $$

while solving any differential equation, what we have to do is to integrate the given differential form and find out the equation of a family of curves which represent a family of curves.

generally to solve homogenous differential equation one can use the following substitution

$$ \frac{y}{x}= t, \hspace{2mm}y=xt $$

the differential of the above equation is used then.

Linear Differential equation

a linear differential equation is a differential equation having some differential coefficient of the dependent variable with respect to the independent variable, and some polynomial the general form of a linear differential equation is-

$$\frac{dy}{dx}+P(x).y=Q(x) $$

here P and Q are functions of “x”, or there may be some constant. We define integrating factor to solve the linear differential equations, Compared to the general form of the Linear differential equation the integrating factor is defined as

$$ I.F= e^{\int P(x).dx} $$

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