DIFFERENTIAL EQUATIONS ENGINEERING MATHEMATICS GATE 2020 STUDY MATERIAL PDF CIVILENGGFORALL

Ordinary Differential Equation
Integrating Factors (I.F.)
Clairaut’s Equation
Particular Integral
Method of Variation of Parameters
First Order Equations (Linear and Non-Linear)
Euler-Cauchy Equations
Partial Differential Equations
Working Procedure to Solve Pp + Qq = R
Applications of Partial Differential Equations
One Dimensional Diffusion Equation
Variable Speed
More Than Two Dimensions
The Second-Order One-Dimensional Wave Equation

ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

An equation which contains independent variable, dependent variables and their derivatives with respect to independent variables, is called differential equation.

ORDINARY DIFFERENTIAL EQUATION

A differential equation which involves only one independent variables is called ordinary differential equation.

Example : dy/dx + xy = 0

And x(dy/dx)²– y(dy/dx) + 1 = 0

Order of Differential equation : It is the order of the highest derivative involving in the equation.

Degree of Differential equation : It is the degree of the highest order derivative involving in the equation, when the equation is free from radicals and fractional powers.

Example: dy/dx + xy = 0 is the equation of first order and first degree

d²y/dx² + x(dy/dx) = 2

is the equation of second order and third degree

Solution of a Differential equation: It is any relation between variables involved which satisfies the differential equation.

General or Complete Solution of a Differential equation: Solution of a differential equation in which number of arbitrary constant is equal to order of the differential equation is called general solution.

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PARTIAL DIFFERENTIAL EQUATIONS

If a dependent variable is a function of two or more independent variables, an equation involving partial differential coefficients is called partial differential equation. The order of a partial differential equation is the same as that of the highest order differential coefficient in it. If z = f(x, y) where x and y are independent variables, then partial differential coefficients δz/ δx, δz/δy will be denoted by p and q respectively.

Formation of Partial Differential Equation : The partial differential equations can be formed either by elimination of arbitrary constants from a relation between x, y, z or by the elimination of arbitrary functions of these variables.

Solutions of Partial Differential Equations

The solution f(x, y, z, a, b) = 0 of a first order partial differential equation, which contains two arbitrary constants is called complete solution or complete integral.
If in this solution, we put b = φ(a) and find envelope of the family of surfaces f(x, y, z, a, φ(a)) = 0, we get a solution involving an arbitrary function
This is called general solution or general integral.  A solution obtained from the complete integral by giving particular values to the arbitrary constant is called particular solution or particular integral.

Equations Solved by Direct Integration : Those equations which contain only one partial derivative can be solved by direct integration. However, in place of the constants of integration, we must use arbitrary functions of the variable kept constant.

Linear Partial Differential Equations of First Order : A differential equation involving first order partial derivatives p and q only is called partial differential equation of the first order. If p and q both occur in the first degree only and are not multiplied together, then it is called a linear partial differential equation of the first order.

Lagrange’s Linear Equation : Partial differential equation of the form Pp + Qq = R, where P, Q and R are functions of x, y, z is the standard form of a linear partial differential equation of the first order and is called Lagrange’s linear equation.