Engineering Mathematics: Differential Calculus

Differential Equation

1.1. Differential Equation: An equation involving a dependent variable and the differential coefficients of the dependent variable with respect to one or more independent variables (Differentials).

1.2. Ordinary Differential equation:  Differential equation, in which differential coefficients are with respect to one independent variable.

1.3. Partial Differential equation: Differential equation, which involves more than one independent variable and differential coefficients with respects to any of these.

1.4 Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations: 

• The first, second and third equations  involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively.

1.5 Degree of a differential equation 

• To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider the following differential equations: 

• We observe that first equation is a polynomial equation in y″′, y″ and y′ with degree 1,  second equation  is a polynomial equation in y′ (not a polynomial in y though) with degree of 2.
• Degree of such differential equations can be defined. But third equation is not a polynomial equation in y′ and degree of such a differential equation can not be defined. 
• By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation. 

NOTE: Order and degree (if defined) of a differential equation are always positive integers. 

1. SOLUTION OF FIRST ORDER AND FIRST-DEGREE DIFFERENTIAL EQUATION

       2.1  Equations with Separable Variable

2.2. Equations Reducible to Equations with Separable Variable

2.3.  Homogeneous Differential Equations

2.4.  Equations Reducible to Homogeneous Equation

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