Solution of a Differential Equation: Overview, Questions, Preparation

This chapter has multiple topics listed below:

Differential Equations Laplace Transform Differential Calculus Ordinary Differential Equations Solution of a Differential Equa...Integrating Factor Partial Derivative Exact Differential Equation Differential Equations Applicat...First Order Differential Equati...

Rachit Kumar SaxenaManager-Editorial

Updated on Jun 29, 2021 12:38 IST

Table of Contents

What is the Solution of differential equations?
Weightage of Solution of a differential equation
Illustrated Examples on Solution of a Differential Equation
FAQs on Solution of a differential equation

What is the Solution of differential equations?

In Mathematics, a differential equation is an equation with derivatives that contains one or more functions. The function derivatives describe the rate of a function's change at a point. It is predominantly used in areas such as astronomy, genetics, engineering, and so on. The primary goal of the differential equation is the analysis of solutions that obey the solutions' equations and properties.

A differential equation is an equation containing one or more terms and derivatives of one variable with respect to the other variable, i.e. the dependent variable (i.e., independent variable)

For eg:
dy/dx = f (x)

Derivatives that are either partial derivatives of ordinary derivatives are found in a differential equation. The derivative represents a rate of change, and the differential equation defines a constantly changing relationship between the quantity in response to the change in another quantity. To find the solution of the derivatives, there are a lot of differential equation formulas.

Weightage of Solution of a differential equation

In Class 12, the calculus part has more weightage, so in the chapter Differential Equation, you will get to learn it in detail. The weightage of this chapter is 7-8 marks in the board exam.

Illustrated Examples on Solution of a Differential Equation

1. What is the Differential Equation Order?
Solution. In the differential equation, the order of the higher-order derivative present is considered the equation order. If the order of the differential equation is 1, the first order is then labelled. If the order of the equation is 2, it is referred to as the second-order, and so on.

2. In differential equations, what is a linear equation?
Solution. A linear differential equation is a differential equation in mathematics that is defined by a linear polynomial in the unknown function and its derivatives, i.e. the form equation.

3. Why are there differential equations that we solve?
Solution. A Differential Equation, on its own, is a fantastic way to express anything, but it is difficult to use. So by converting the Differential Equation into a simpler equation without the differential parts, we attempt to solve them, so we can do equations, make diagrams, forecast the future, and so on.

FAQs on Solution of a differential equation

Q: What are differential equations and their forms?

A: All differential equations can be placed into two types: ordinary differential equations and partial equations of differentials. A differential equation that includes partial derivatives is a partial differential equation. A differential equation that does not include partial derivatives is an ordinary differential equation.

Q: What is the general solution to the equation of differentials?

A: In terms of the independent one(s) that satisfies the relation, a solution of a differential equation is an expression for the dependent variable. All potential solutions are found in the general approach that normally contains arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.)

Q: Is that the calculus of differential equations?

A: The basic theorem of calculus, which says that differentiation is the opposite method of integration, links differential calculus, and integral calculus. Differential equations are called equations involving derivatives and are important in the explanation of natural phenomena.

Q: Why are there differential equations we use?

A: Differential equations are used to model the behaviour of complex systems in biology and economics. Along with the sciences where the calculations originated and where the results find use, the mathematical theory of differential equations first emerged.

Q: When, in real life, are differential equations used?

A: In real life, ordinary differential equations are used to measure the acceleration or flow of energy, the rotation of an object to and fro like a pendulum, to illustrate the laws of thermodynamics. They are also used in medical terminology to validate the growth of diseases in graphical representation.