# RS Aggarwal Class 12 Chapter 21 Solutions (Linear Differential Equations)

RS Aggarwal Solutions for Class 12 Maths Chapter 21 ‘Linear Differential Equations’ have been prepared to help you ace your board exam and important entrance exams such as JEE, NEET etc. In this chapter, you will learn about the conditions that are essential to define a linear equation. Also, you will become proficient in finding the solutions of all kinds of linear equations, once you get clarity in evaluating the solutions of Linear Differential Equations. This chapter of RS Aggarwal Solutions is very crucial to get a complete understanding of Differential Equations and carries a significant weightage in most school exams.

There are a total of 20 questions included in one exercise. The exercises in RS Aggarwal Solutions for Class 12 Maths Chapter 21 Linear Differential Equations will help you to practice the most frequently asked exam questions in a rigorous fashion. Thus, referring to Instasolv for solving the questions of the exercises will help you boost your immunity against the silly mistakes committed in the exam as usual.

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## Summary of Important Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 21 – Linear Differential Equations

**Introduction to Linear Differential Equations:**

We may express the differential equation that is linear in nature and has an order n, with the dependent and independent variable y and x respectively, in the following form:

Such that a_{0 }is not equal to 0.

**Properties of the Linear Differential Equations:**

There are certain features possessed by linear differential equations which are described as follows:

- A given function y and the derivatives of y can be evaluated in the differential equation only up to the first degree.
- There is no existence of the product of the function y and its supposed derivatives.
- The transcendental functions such as logarithmic or trigonometric functions of the given function y or its derivatives are totally absent.

**Linear Differential Equation of the First Order:**

Any linear differential equation which includes just the function y and the first derivative of y is termed as a Linear Differential Equation of the First Order.

It can be denoted as

dy/dx + P(x).y= Q(x)

P(x), Q(x)Continuous Functions

To solve these equations of the first order, we must reduce either of the two functions P(x) or Q(x) equivalent to 0, after which it will become feasible to use the variable separable technique. This is how you will be able to evaluate the solutions of linear differential equations effortlessly.

**Bernoulli’s Equation as Linear Differential Equation**

The Bernoulli’s equations can be denoted as follows

y’+p(x)y=g(x)y^{a}

Such that a is a real number. For a= 1 or a= 0, the equation can be simply solved as a linear equation is solved;

Whereas if a has some other value, we can employ the steps as mentioned below:

- The first step is to assume a new dependent variable u(x) such that

u(x)=[y^{1-a}(x)]

- Now, find the derivation of u which will be given as

u’=(1-a)y^{-a}y’

- You can now substitute the value of y’ from the given differential equation in the above equation to obtain

u’=(1-a)(g-pu)

- Rearranging the above equation, we will obtain:

u’+(1-a)pu=(1-a)g

- You can now solve it as a linear differential equation assuming P(x)= (1-a)pu and Q(x)=(1-a)g

**Major Highlights of Linear Differential Equations:**

- Product rule of Differentiation:

Thus we can solve a differential equation by substituting the value of (x)in the value of y for the product rule of differentiation.

### Exercise Wise discussion of RS Aggarwal Solutions for Class 12 Maths Chapter 21 – Linear Differential Equations

- The exercise solutions in RS Aggarwal Class 12 Chapter 21 are so designed by the experts such that you will be able to grasp the gist of the topic and its applications through solving these questions rigorously.
- The exercises in this chapter will give you a substantial opportunity to enhance your analytical skills to be able to begin an answer with the right approach.
- There are all kinds of questions which will cover objective answers and subjective answers both.

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- We have covered all the questions in a detailed step by step fashion ensuring the quality of our answers by making those solutions be checked at various stages by different maths experts.
- You will find the RS Aggarwal Class 12 Solutions provided at Instasolv truly reliable for your homework and quick revision before your school exam.