1.

(i) An equation containing an independent variable, dependent variable and differential coefficients of the dependent variable with respect to the independent variable is called a differential equation.
(ii) The order of a differential equation is the order of the highest order derivative appearing in the equation.
The order of a differential equation is a positive integer.
(iii) The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions.

In other words, the degree of a differential equation is the power of the highest order derivative occurring in a differential equation when it is written as a polynomial in differential coefficients.

2. A differential equation is a linear differential equation if it is expressible in the form

\displaystyle P_0 \frac{d^n y}{dx^n} + P_1 \frac{d^{,n-1} y}{dx^{,n-1}} + P_2 \frac{d^{,n-2} y}{dx^{,n-2}} + \cdots + P_{n-1} \frac{dy}{dx} + P_n y = Q

where \displaystyle P_0, P_1, P_2, \ldots , P_{n-1}, P_n and Q are either constants or functions of the independent variable x.

Thus, if a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no products of these, and also the coefficients of the various terms are either constants or functions of the independent variable, then it is said to be a linear differential equation.

Otherwise, it is a non-linear differential equation. It follows from the above definition that a differential equation will be non-linear if: (i) its degree is more than one.
(ii) any of the differential coefficients has exponent more than one.
(iii) exponent of the dependent variable is more than one.
(iv) products containing dependent variable and its differential coefficients are present.

3. The solution of a differential equation is a relation between the variables involved which satisfies the differential equation. Such a relation and the derivatives obtained therefrom, when substituted in the differential equation, makes left-hand and right-hand sides identically equal. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution of the differential equation.

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

4. A differential equation is said to be in the variable separable form if it is expressible in the form

\displaystyle f(x),dx = g(y),dy

The solution of this equation is given by

\displaystyle \int f(x),dx = \int g(y),dy + C

where C is a constant.

5. A differential equation of the form \displaystyle \frac{dy}{dx} = f(ax + by + c) can be reduced to variable separable form by the substitution \displaystyle ax + by + c = v.

6. If a first-order first-degree differential equation is expressible in the form

\displaystyle \frac{dy}{dx} = \frac{f(x,y)}{g(x,y)}

where f(x,y) and g(x,y) are homogeneous functions of the same degree, then it is called a homogeneous differential equation.

Such type of equations can be reduced to variable separable form by the substitution

\displaystyle y = vx \quad \text{or} \quad x = vy.

7. If a differential equation is expressible in the form

\displaystyle \frac{dy}{dx} + P y = Q

where P and Q are functions of x, then it is called a linear differential equation. The solution of this equation is given by

\displaystyle y,e^{\int P,dx} = \int \left( Q,e^{\int P,dx} \right) dx + C

Sometimes a linear differential equation is in the form

\displaystyle \frac{dx}{dy} + R x = S

where R and S are functions of y. The solution of this equation is given by

\displaystyle x,e^{\int R,dy} = \int \left( S,e^{\int R,dy} \right) dy + C

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