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**Definition**: An equation involving the independent variable x(say), dependent variable y(say) and the differential coefficient of dependent variable w.r.t independent variable i.e dy/dx, d^{2}y/d^{2}x,…. etc is called differential

equation.

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

**Degree** of a differential equation is the degree of highest order derivative occurring in the differential equation when the differential coefficients are made free from radicals, fractions and it is written as a polynomial in differential coefficients.

e.g(d^{2}y/d^{2}x)^{3}+sin(dy/dx) = 0 here order is 2 but this differential equation can’t be written in the form of polynomial in differential coefficient Hence its degree not defined.

**Linear and Nonlinear Differential Equations:** A differential equation in which the dependent variable and its derivatives occur only in the first degree and are not multiplied together, is called a linear differential equation otherwise it is non-linear.

**“Formation of differential Equation”** To form a DE from a given equation in x and y containing arbitrary constants (parameters) –

**“Initial value problem(IVP)** is one in which some initial conditions are given to solve a DE”

1. Differentiate the given equation as many times as the number of arbitrary constants involved in it.

2. Eliminate the arbitrary constant from the equations of y, y’, y’’ etc.

**Solution of Differential Equations-**

1. Variable separable form

2. Homogenous Equations

3. Linear Differential Equations

**VARIABLE SEPARABLE FORM** If in the equation, it is possible to get all terms containing x and dx to one side and all the terms containing y and dy to the other, the variables are said to be separable,

**Procedure to solve:**

Consider the equation dy/dx= X.Y, where X is a function of x only and Y is function of y only.

(i) Put the equation in the form 1/Y ,dy=X.dx.

(ii) Integrating both the sides we get

**Homogeneous Differential Equations**

A differential equation which can be expressed in the form dy/dx=f(x,y) or dx/dy=g(x,y) where, f (x, y) and g(x, y) are homogenous functions of degree zero is called a homogeneous differential equation

**Steps for Solving a Homogeneous Differential Equation**

1. Rewrite the differential in homogeneous form

2. Make the substitution y = vx orx = vy where v is a variable.

3. Substitute to rewrite the differential equation in terms of v and x or v and y only

4. Follow the steps for solving separable differential equations.

5. Re-substitute v = y/x or v = x / y in the final solution.

**Linear Differential Equation:** A first-order linear differential equation can be written in the form dy/dx+PY=Q)

where P and Q are constants or function of x only or dx/dy+Px = Qwhere P and Q are constants or function of

y only.

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