RS Aggarwal Class 12 Solutions Chapter-19 Differential Equations with Variable Separable

Differential equations are used in a variety of disciplines, including applied science, mathematics, and engineering. Apart from technical applications, they are also used to solve a variety of real-life concerns. Let's look at some real-time applications of differential equations:

• Differential equations are used to describe the growth and decay of various exponential functions.

• They're also used to show how a return on investment has changed over time.

• They are applied in the field of medical science to represent cancer growth and illness spread in the body.

• These equations can also be used to explain the motion of waves or a pendulum.

• It can also be used to describe the movement of electricity.

• They support economists in determining the best investment strategy.

• Other technical uses include: ­ heat conduction analysis, and in physics, it may be used to explain wave motion.

Types of Differential Equations

Differential equations are classified based on a variety of variables including order, type, linearity, and homogeneity. 

Based on The Type of Variables, The Differential Equation Types are:

• Ordinary Differential Equations: Ordinary differential equations are those with only one independent variable.

• Partial Differential Equations: These are the types of differential equations in which the dependent variable is influenced by two or more independent variables.

Differential Equation types are classified based on the order of the equations and those are:

• First order differential equation: The derivative's highest power is one. 

• Second order differential equation: The derivative's highest power is two.

• N^th order differential equation: Any number 'n' can be the largest power of the derivative.

The following is a list of differential equations based on their linearity:

• Linear differential equations: These are differential equations where the variable's power is always one.

• Nonlinear differential equations: Differential equations with any number other than 1 as the power of the variables in the equation.

Based on Homogeneity, the Differential Equation types are:

• Homogeneous differential equations

• Non homogeneous differential equations

The Solution of Differential Equations

The general solution of the deferential solution is the correlation between the variables x and y that is obtained when the derivatives are removed, and the relation contains arbitrary constants to reflect the order of an equation. The first order differential equation solution has one arbitrary constant, whereas the second order differential equation solution contains two arbitrary constants. When arbitrary constants are given specified values, the general solution of the differential equation is obtained. Some common forms are given to derive the general solution to the first order differential equation of first degree, as follows:

• Homogeneous Equation

• Non homogeneous differential equations

• Linear Equation 

• Exact Differential Equation

• Variable desperate method

Fundamental principles of differential equations

• The derivatives of the dependent variable with regard to only one independent variable are used in the Ordinary Differential Equation. F(x, y, y',...., yn-1) = yn is the formula for ordinary differential equations.

• There are derivatives for more than one independent variable in a partial differential equation. 

• The partial differential equation has the formula ∂u/∂x (x,y) = 0

• The highest order derivative of the dependent variable with respect to the independent variable determines the order of a differential equation.

• A first-order linear differential equation has the form dy/dx +Py Q, with P and Q being constants or functions of just x.

• A function that satisfies the specified differential equation is referred to as a solution.

Conclusion:

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