Mathematics Revision Notes of Differential Equations for NDA

Differential Equatitions

An equation that involves independent variables, dependent variables, derivatives of the dependent variables w.r.t. independent variables and constant, is called a differential, equation.

Types of Differential Equations:

Ordinary Differential Equation Ordinary differential equations are those equations which contain only one independent variable.

In particular, an equation involving only one independent variable (say x), dependent variable (say `y`) and the differential coefficients `(dy)/(dx), (d^2y)/(dx^2)` called ordinary differential equation.

Partial Differential Equation : Partial differential equations are those equations which contain two or more independent variables.

Order and Degree of a Differential Equation

The order of a differential equation is the order of the highest derivative occurring in the differential equation.

By the degree of differential equation, when it is polynomial equation in derivatives, then the greatest exponent (positive integral index) of the highest order derivative, occuring in the differential equation.

The order and degree of a differential equation is always a positive integer.

The degree of the differential equation is found only when the equation is polynomial in derivatives.

Solution of Differential Equation

A primitive or solution of a differential equation is a functional relation between `x` and `y` which is free from derivatives and this relation on substitution satisfies the differential equation.

There are two types of solutions of a differential equation :

General Solution

A general solution of a differential equation is a relation between the variables (not involving the derivatives) which contains the same number of the arbitrary constants as the order of the differential equation

Particular Solution

Particular solution of the differential equation obtained from the general solution by assigning particular values to the arbitrary constant in the general solution.

Formation of Differential Equation whose General Solution is Given

Formulating differential equation that represent the family of given curves, means finding a differential equation whose solution is the given equation. Suppose an equation contains n arbitrary constants, then differentiate the given equation n times to obtain n equations. The
equation so obtained by eliminating arbitrary constant is the differential equation of order n for the family of given curves.

Solution of Ordinary Differential Equation of the First Order and First Degree

An ordinary differential equation of the first order and first degree is of the form

`M+ N (dy)/(dx) =0` or `Mdx+ Ndy=0`

where, `M` and `N` are functions of `x` and `y` or constant. The general solution of such equation will contain only one
arbitrary constant.

Differential equation of first degree and first order can be classified into following types

Variables Separable Differential Equation

If the differential equation can be put in the form `f(x)dx + phi (y)dy = 0` will be termed as variable separable. Such equation will be solved by integrating and adding arbitary constant in side.

Hence, complete general solution of `f(x)dx + phi(y)dy = 0` will be `int f(x)dx + int phi(y) dy =C`

where, `C` is an arbitrary constant or constant of integration.

Reducible to Variable Separable Form

Differential equations of the form `(dy)/(dx) =f(ax+by+ c)` can be reduced to variable separable form by the substitution
`ax+ bv + c = z`

`:. a+b (dy)/(dx) =(dz)/(dx)`

`=> ((dz)/(dx) -a) 1/b = f(z)`

`(dz)/(dx)= a+ b f(z)`

Homogeneous Differential Equation of First Order and First Degree

A differential equation of the form `(dy)/(dx) =(f(x,y))/(phi (x,y))`

where, `f(x, y)` and `phi(x, y)` are both homogeneous functions of `x` and `y` of the same degree, is called a homogeneous
differential equation.

The solution of th's equation is obtained by substituting, `y = vx` or `x = vy` according as `f(x, y)` and `phi(x, y)` is converted
into function of `(y/x)` or `(x/y)`.

This substitution reduces the given equation to the variable separable form.

Reducible to Variable Separable Form

(i) Consider the differential equation of the form `(dy)/(dx) =(ax+by+c)/(a'x +b'y +c')` where `a/(a') ne b/(b')`

This equation is not homogeneous. In order to reduced this equation to the homogeneous form substitute `x= X + h, y = Y + k`, where h and `k`, are constants which are to be determined.

`=> (dy)/(dx) =(dY)/(dX)`

and the above equation becomes

`(dY)/(dX) =(a (X+h) +b(Y+k)+c)/(a' (X+h) +b'(Y+k)+c')`................(i)

`(dY)/(dX)= ((aX+bY)+ah+bk+c)/((a'X+b'Y)+ a'h+b'k+c')`...................(ii)

Now, `h` and `k` will be chosen such that

`ah+bk+c=0`

`a'h +b'k +c'=0`

`h/(bc' b'c)= k/(ca' - c'a)`

`= 1/(ab' -a'b)`

For these values of `h` and `k` in Eq. (ii) reduces to

`(dY)/(dX) =(aX+bY)/(a'X +b'Y)`

which is a homogeneous equation in `X, Y` and can be solved by the substitution `Y = vX`.

`(dY)/(dX) = v+ X * (dV)/(dX)`

Replacing, `X` and `Y` in the solution so obtained by `x - h` and `y - k`, respectively, we can obtain the required solution in term of `x` and `y`.

(ii) If `(dy)/(dx) =(ax+by+c)/(a'x+b'y + c')` and `a/(a') = b/(b') =m`

Then, `(dy)/(dx) =(m(a'x +b'y)+c)/(a'x +b'y +c')`

where, `m` is any number.

In such case substitute `a' x + b' y = v`

So, that `a' +b' (dy)/(dx) =(dv)/(dx)`

Transform the differential equation of the form

`1/(b') ((dv)/(dx) -a') =(mv+c)/(v+c')` we get

`(dv)/(dx) = a' +b' ((mv+c)/(v+c'))`

which is a differential equation in variable separable form and it can easily be solved.


Linear Differential Equation of First Order

A differential equation of the form `(dy)/(dx)+Py=Q`

where, `P` and `Q` are either constants or functions of `x` (and not of `y`) is said to be linear differential equation of first
order.

To obtain the solution of this equation, we first obtain an integrating factor (IF) as

`IF= e^(int Pdx)`

The solution of the given equation is given by

`y*(IF)= int Q (IF) dx + C`

NOTE : If the equation is given by `(dx)/(dy) +Px =Q` where `P` and `Q` are functions of `y`(not of `x`), then `IF= e^(int Pdy)` and solution will be `x * (IF)= int Q (IF)dy+ C`.

Before solving linear differential equation, make the coefficient of `(dy)/(dx) ` or `(dx)/(dy)` equal to `1`

Extended form of Linear Equations

(i) Bernoulli's Equation :

An equation of the form `(dy)/(dx) +Py = Q y^n` where `P` and `Q` are functions of `x` alone or constant and `n` is
constant, other than `0` and `1`, is called a Bernoulli's equation.

This equation can be reduced to the linear form by dividing `y^n`, we get

`y^(-n) (dy)/(dx) +P * y^(n+1) =Q`

Now, put `y^(-n+1) = v`, so that

`(-n+1) y^(-n) (dy)/(dx) =(dv)/(dx)`

`(dv)/(dx) +(1-n) Pv= (1-n) Q`

which is a linear differential equation.

(ii) If the equation of the form

`(dy)/(dx) +P phi (y) = Q psi(y)`

where, `P` and `Q` are functions of `x` alone or constant.

Dividing the given equation by `psi(y)`, we get

Now Put `(phi(y))/(psi(y))=v` so that `d/(dx) { (theta (y))/(psi(y))}=(dv)/(dx)`

or `(dy)/(dx) =k * 1/(psi (y)) *(dy)/(dx)` where `k` is constant.

We get `(dv)/(dx) +k Pv = kQ`

which is a linear differential equation.

 
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