Differential Equation

An equation containing an independent variable, dependent variable and differential
coefficients of dependent variable w.r.to independent variable is called a differential equation.
Order
The order of a differential equation is the order of the highest order of derivative appearing in the equation.

Degree
The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions.

Formation of differential equation
Formulating a differential equation from a given equation representing a family of curves means finding a differential equation whose solution is the given equation. If an equation representing a family of curves contains, n orbitrary constants, then we differentiate the given equation 'n' times to obtain n- more equation. Using all these (n+1) equation, we eliminate the constants. The equation so obtained is the differential equation of order 'n' for the family of given curves.

Solution of a differential equation
The solution of a differential equation is a relation between the variables involved which satisfies the differential equation.

General Solution
The solution which contains as many as arbitrary constants as the order of the differential equation is called the general solution of the differential equation.

Particular solution
Solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equation is called a particular solution.

Method of solving a first order first degree differential equation:

Type-1
Differential equations of the type $\frac{dy}{dx}$ = f(x)

$\frac{dy}{dx}$ = f(x) $\Rightarrow$ dy = f(x).dx then integrating both sides.

Type-ll
Differential equations of the type $\frac{dy}{dx}$ = f(y)

$\frac{dy}{dx}$ = f(y) $\Rightarrow \frac{dy}{f (y)}$ =dx, then integrating both sides.

Type -lll Variable separable
If the differential equation can be put in the form f(x) dx = g(y).dy, we say that the variables are separable and such equations can be solved by integrating on both sides.

Type- IV
Equations reducible to variable separable
Differential equations of the form $\frac{dy}{dx}$ = f(ax + by +c) can be reduced to variable separable form by the substitution ax+by+c= v

Type-V Homogeneous differential equations
If a first order first degree differential equations is expressible in the form $\frac{dy}{dx}$ = $\frac{f (x, y)}{g (x, y)}$ where f(x,y) and g(x,y) are homogeneous differential equations.
Such type of equations can be reduced to variable separable form by the substitution y = vx.

$∴$ $\frac{dy}{dx}$ = V + x. $\frac{dv}{dx}$

Type- VI Equations reducible to homogeneous form
If the differential equation is of the form

$\frac{dy}{dx}$ = $\frac{a_{1} x + b_{1} y + c_{1}}{a_{2} x + b_{2} y + c_{2}}$
then putting x = x + h,

y= y+k $\Rightarrow$ $\frac{dy}{dx} = \frac{dy}{dx} . ∴$ Given differential equation becomes $\frac{dy}{dx}$ = $\frac{a_{1} X + b_{1} Y + {(a}_{1} h + b_{1} k + c_{1})}{a_{2} X + b_{2} Y + {(a}_{2} h + b_{2} k + c_{2})}$

Choose h and k such that $a_{1} h + b_{1} k + c_{1} = 0$

$a_{2} h + b_{2} k + c_{2}$ = 0 for these values of h and k above equation reduces to homogeneous.

Notes
If $a_{1}$ $b_{1}$ - $a_{2}$ $b_{1}$ = 0 or $\frac{a_{2}}{a_{1}}$ = $\frac{b_{2}}{b_{1}}$ =k (say)

$∴ a_{2}$ = ${ka}_{1}$ , $b_{2}$ = ${kb}_{1}$
Then the given equation can be written as $\frac{dy}{dx}$ = $\frac{a_{1} x + b_{1} x + c_{1}}{k (a_{1} x + b_{1} y) + c_{2}}$ and putting v=
$a_{1} x + b_{1}$ y

Linear Differential Equations

Type-VII
A differential equation is of the form $\frac{dy}{dx}$ +Py- Q where P and Q are functions of x ( or constants)
Then multiplying both sides by $e^{\int Pdx}$ which is known as the integrating factor (I.F).
To solve the above equation multiplying both sides of the given equation by $e^{\int Pdx}$ and integrating.

Type-VIII
Linear differential equations of the form $\frac{dy}{dx}$ +Rx= S. In this case the integrating factor is ${}_{e}\int$ Rdy, then multiplying both sides by I.F and integrating.

Type- IX Equations reducible to linear form ( Bernoulli's differential equation)
The differential equation of the type $\frac{dy}{dx}$ +Py= Q. $y^{n}$ . where P and Q are constants or functions of 'x' alone, can be reduced to the linear form by dividing with $y^{n}$ and then putting $y^{- n + 1}$ = z.