# RS Aggarwal Class 12 Chapter 19 Solutions (Differential Equations with Variable Separable)

RS Aggarwal Solutions for Class 12 Chapter 19 ‘Differential Equations with Variable Separable’ are prepared to guide you in solving the problems in this chapter easily. Differential Equations with Variable Separable are the most recurring kind of Differential Calculus with a wide range of applications from physical sciences, chemical and medical sciences to social science. The topics covered in this chapter of RS Aggrawal Solutions include an introduction to differential calculus and the algorithmic steps to evaluate general solutions for given differential calculus.

There are 28 questions in a total of 2 exercises with thorough coverage of almost all types of functions. The questions are of a wide variety and are mentioned in an order of increasing level of difficulty. Solving the exercises of RS Aggarwal Class 12 Solutions for Chapter 19 ‘Differential Equations with Variable Separable’ will make you proficient in managing your time in the environment of increased pressure during an exam. The questions are of varying categories ensuring ample opportunity for you to practice the topics rigorously.

The team of experts at Instasolv use research-based data to address the most frequently occurring doubts. We have written the answers in such a way that you will be able to grasp the concepts in a step by step manner. Referring to the RS Aggarwal Solutions for Class 12 Chapter19 at Instasolv will enhance your analytical skills and approachability towards Differential Equations problems. The experts at Instasolv use simple language and adhere to the trending CBSE pattern.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 19 – Differential Equations with Variable Separable

**Introduction:**

A separable differential equation is the most usual kind of differential calculus equation that is easy to calculate. These are of the form dy/dx = f(x) g(y) and are known as separable because we can bring x and y to separate sides of the equations.

Then, integrating both sides gives y as a function of x, solving the differential equation. Separable differential equations are useful because they can be used to understand the rates of chemical reactions, the growth of populations, the movement of projectiles, and many other physical systems.

**Differential Calculus with Variables Separable**

Many practically useful first-order differential equations can be reduced, by purely algebraic manipulations, to the form –

g(y) y′ = f(x)

OR

g(y) dy = f(x) dx

where f and g are continuous functions This is known as the Variables Separable form of the differential equation. The name must be clear from the format of the equation since all of the terms with the independent variable x are on one side, while all of the terms with the dependent variable y are on the other side. Such an equation can be easily solved for its solution in the following way –

Integrate on both sides with respect to x

∫ g(y) y′ dx = ∫ f(x) dx + c

since y’ dx is same as dy,

∫ g(y) dy = ∫ f(x) dx + c

**Highlights of Differential Equations with Variable Separable:**

Radioactive Decay:

As we have studied in physics about the phenomenon of radioactive decay is governed by the differential equation –

where N is the number of radioactive atoms existing in the sample at time t,

k is proportionality constant (the radioactive decay constant).

This differential equation is clearly a variables-separable equation. Let us now solve it for the general solution –

where c is the arbitrary constant.

We can find the constant c by using the initial conditions of the system. Put t = 0, and N = N_{0} (the initial number of radioactive atoms in the system) to get the value of c.

ln(N_{0}) = −k.0 + c

c = ln(N_{0})

Hence the particular solution will be –

ln(N) = −kt + ln(N_{0})

N = N_{0}e^{−kt}

**Exercise Discussion of RS Aggarwal Solutions for Class 12 Chapter 19 – Differential Equations with Variable Separable**

- The exercise solutions of 19A include the beginner level questions which need you to write short answers and are more objective in nature.
- The solutions in exercise 19B are one level up in this topic and will prepare you to approach difficult questions effortlessly.
- The exercises are arranged in a manner such that the easier questions will be the first ones you will get to solve and the difficulty level increases gradually as you progress.
- These solutions will help you to arrive at the answers to objective questions using short tricks and these solutions are detailed enough to be used as answers in the long subjective answer-type questions.
- Practising Instasolv’s solutions for RS Aggarwal Class 12 Chapter 19 will help you to increase your marks significantly.
- These questions are sufficient for revision before exams complemented with your course textbook.

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- Instasolv experts give special emphasis on helping you grab the gist of the chapter to make it an interesting subject for you and make you aware of its applications.
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