Any equation, in which derivatives or differentials of one or more dependent variables with one or more independent variables are involved, is called differential equation and is denoted by
etc. For eg.
} \frac{dy}{dx}= \text{sinx} "\text{(i)} \frac{dy}{dx}= \text{sinx}")
}\frac{d^2y}{dx^2} + \5 \frac{dy}{dx}+7y = 3 "\text{(ii)} \frac{d^2y}{dx^2} + \5 \frac{dy}{dx}+7y = 3")
,
y is dependent variable and x is independent variable.
a) Types of differential equation:-
A differential equation is said to e ordinary diffn if it involves derivatives of dependent variables with respect to single independent variable. For eg.
, 
b) Partial differential equation:-
A differential equation is said to be partial differential equation if it involves derivative of dependent variables with respect to two or more independent variables. For eg.
} \frac{\partial^2u }{\partial x^2} + \frac{\partial^2u }{\partial y^2} = \text{0},{u} = \text{f}\left (x,y \right ) "i) \frac{\partial^2u }{\partial x^2} + \frac{\partial^2u }{\partial y^2} = \text{0},{u} = \text{f}\left (x,y \right )")
} \frac{x\partial z }{\partial x} + \frac{y \partial z }{\partial y} = \text{5z},\text{z}= \text{f}\left (x,y \right ) "\text{(ii)} \frac{x\partial z }{\partial x} + \frac{y \partial z }{\partial y} = \text{5z},\text{z}= \text{f}\left (x,y \right )")
Order of differential equation:-
The order of differential equation is the order of highest derivative appearing in it.For eg.
} \frac{\partial ^4y}{\partial x^4}+ \frac{\partial ^2y}{\partial x^2} + \left ( \frac{\partial y}{\partial x} \right )^2 = 5 \to \text{it is the order of 4.} "\text{i)} \frac{\partial ^4y}{\partial x^4}+ \frac{\partial ^2y}{\partial x^2} + \left ( \frac{\partial y}{\partial x} \right )^2 = 5 \to \text{it is the order of 4.}")
} \frac{\partial ^2y}{\partial x^2}+ \text{3} \frac{\partial y}{\partial x} =2 \to \text{it is the order of 2.} "\text{ii)} \frac{\partial ^2y}{\partial x^2}+ \text{3} \frac{\partial y}{\partial x} =2 \to \text{it is the order of 2.}")
} \sqrt{1 - x^2} \partial x + \sqrt{1 - y^2} \partial y = 0 \to \text{it is the order of 1.} "\text{iii)} \sqrt{1 - x^2} \partial x + \sqrt{1 - y^2} \partial y = 0 \to \text{it is the order of 1.}")
Degree of differential equation:-
The degree of differential equation is the degree of highest derivative involving the equation after removing radicals or fractions. For eg.
} 2\frac{\partial ^2y}{\partial x^2} - 5 \frac{\partial y}{\partial x} + 7 = 0 \to \text{it is} \2^{nd} \text {order and} \1^{st} \text{degree}. "\text{i)} 2\frac{\partial ^2y}{\partial x^2} - 5 \frac{\partial y}{\partial x} + 7 = 0 \to \text{it is} \2^{nd} \text {order and} \1^{st} \text{degree}.")
} \frac{\partial ^2y}{\partial x^2} = \left [ 1 + \left(\frac{\partial y}{\partial x} \right )^2 \right ]^\frac{3}{2} "\text{ii)} \frac{\partial ^2y}{\partial x^2} = \left [ 1 + \left(\frac{\partial y}{\partial x} \right )^2 \right ]^\frac{3}{2}")
becomes
^2 = \left(1 + \left(\frac{\partial y}{\partial x} \right )^2 \right )^3 \to \text {it is} \2^{nd} \text {degree and } \2^{nd} \text {order.} "\left(\frac{\partial ^2y}{\partial x^2} \right )^2 = \left(1 + \left(\frac{\partial y}{\partial x} \right )^2 \right )^3 \to \text {it is} \2^{nd} \text {degree and } \2^{nd} \text {order.}")
Solution of the Differential eqn:-y is dependent variable and x is independent variable.
a) Types of differential equation:-
A differential equation is said to e ordinary diffn if it involves derivatives of dependent variables with respect to single independent variable. For eg.
b) Partial differential equation:-
A differential equation is said to be partial differential equation if it involves derivative of dependent variables with respect to two or more independent variables. For eg.
Order of differential equation:-
The order of differential equation is the order of highest derivative appearing in it.For eg.
Degree of differential equation:-
The degree of differential equation is the degree of highest derivative involving the equation after removing radicals or fractions. For eg.
becomes
Solution of the differential eqn is the function relation between the variables involved in differential.
There are two types of solution and they are:-
i) General Solution which gives family of curves and
ii) Particular Solution which gives a particular curves.
Linear differential eqn:-
A differential equation is said to be linear differential equation if dependent variable & derivative involved in it appear only in first degree.
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Hi! I am Nirazz and live in 
Thanks for giving such useful notes on advance calculus.....
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