Equations and General Solution of a DE

There are many different methods of solving a differential equation (or DE). The two particular types of solutions are general and particular solutions, which use some integration steps to find the solution. Here, we will be focusing on – General solution of a DE.

The general solution of a DE is the most basic form of the solution that does not regard importance to any initial condition. The general solution of a DE correlates with the variables x and y, which are received after the removal of derivatives (i.e., after integration) where arbitrary constants representing the order of an equation are included in the relation.

Introduction to the Concept of Differential Calculus and Differential Equations

Here, we will be discussing concepts related to differential calculus, differentiation, and differential equations.

The Concept of Differential Calculus and Differentiation

• Differential calculus is the branch of mathematics that deals with studying the rate of change of a quantity with respect to another and it is a subfield of calculus

• The primary objective of differential calculus is to study derivatives of a function, differential equations, and their applications in real life

• The function’s derivative describes the rate of change of a function near a chosen input value

• The process to find a derivative of a function is termed differentiation

• The derivative of a function at a point is geometrically defined as the area of the slope of a tangent to the function at that given point in a graph

• The process of differentiation is reversed to the process of integration

• The concept of differentiation is applicable in many quantitative processes

• It is applied in various concepts of physics, chemistry and mathematics, such as – finding the acceleration of a moving body by differentiating its velocity with respect to time; where velocity is given by differentiating the displacement with respect to the time taken by the moving body, in Newton’s second law of motion, finding the rate of a chemical reaction, finding the maxima and minima of a given function and others

The Concept of Differential Equations

• Equations involving derivatives of a dependent variable with respect to an independent variable are known as differential equations

• For example, x (dy/dx) – y = 0 is a differential equation

• An ordinary differential equation is defined as that differential equation that involves the derivatives of a dependent variable concerning only a single independent variable

• For example, (d2y/dx2) + 2 (dy/dx)3 = 0 is an ordinary differential equation

• Differential equations that involve derivatives of a dependent function concerning more than a single independent variable are known as partial differential equations

• For example, (∂2u/∂t2) – c2 (∂2u/∂x2) = 0 is a partial differential equation

• Differential equations are classified according to their order

• (dy/dx) and (d2y/dx2) are sometimes expressed as y’ and y’’ respectively

Introduction to the Concepts of the Particular and General Solution of a DE

• There are many different methods of solving a differential equation (or DE)

• The two particular types of solutions are general and particular solutions, which use some integration steps to find the solution

• Here, we will be focusing on – General solution of a DE

Example: Consider a function y = Ø (x) = a cos (x + b)

Here, function Ø consists of two parameters (i.e., arbitrary constants) – a and b, and hence, it gives the general solution of a DE. In contrast, consider a function y = Ø1 (x) = 4 cos (x + π). Here, a = 4 and b = π. We can see that function Ø1 does not consist of arbitrary values but instead contains a particular value of a and b, and hence, it gives the particular solution of a DE.

The General Solution of a DE

The general solution of a DE is the most basic form of the solution that does not regard importance to any initial condition.

The general solution of a DE correlates with the variables x and y, which are received after the removal of derivatives (i.e., after integration) where arbitrary constants representing the order of an equation are included in the relation.

For example, for a function y = Ø (x) = a cos (x + b), function Ø consists of two arbitrary constants – a and b, and hence, it gives the general solution of a DE.

The General Solution of a DE according to its Order

• The general solution of a DE geometrically represents the n-parameter family of curves, where n is the order of that differential equation

• Hence, the general solution of a DE with first order will have one arbitrary constant introduced after integration, with second-order will have two arbitrary constants, and so on

Formation of a Differential Equation from the General Solution of a DE

• One can find the differential equation for a function whose general solution is given

• For example, the equation:

x2 + y2 + 2x – 4y + 4 = 0 is the general solution of a DE representing a circle with centre at (-1, 2) and radius of 1 unit.

On differentiating the above function with respect to x, we obtain

(dy/dx) = {(x + 1) / (2 – y)}, where y ≠ 2.

The above equation is a differential equation derived from the general solution.

Conclusion

From this article on Calculus – Differential Equations – General Solution of a DE – students have been made familiar with the concepts of differential calculus and their applications, differential equations in calculus, and particular and general solutions of a DE. 

They can now grasp the concepts of the general solution of a DE, general solution according to the order of a DE, and forming differential equations from the general solution of a DE.