# RS Aggarwal Class 12 Chapter 20 Solutions (Homogeneous Differential Equations)

RS Aggarwal Solutions for Class 12 Maths Chapter 20 ‘Homogeneous Differential Equations’** **is prepared to provide you assistance for solving the exercise in this chapter. The Chapter- Homogeneous Differential Equations of RS Aggarwal Solutions include a detailed introduction of the topic. You will also get to gather knowledge about homogeneous differential equations of the first order and how to convert a differential equation into a homogeneous equation and the conditions essential for the same. This chapter is of high importance, also for your higher studies which is why you must practice as many questions as possible to get empowered in pursuing your dream career.

This chapter contains 1 exercise with 15 questions in it. You will get ample opportunity to get answers to your curiosity arising with the help of the different variety of problems and solutions written by the expert maths faculty of Instasolv. The symbols used in the solutions to denote the formulae are in an authentic format making RS Aggarwal Class 12 Maths Solutions for Chapter 20 a reliable source of reference. These questions will also help you become lucid in approaching the maths problem related to the homogeneous differential equations.

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## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 20 – Homogeneous Differential Equations

**Introduction:**

In this chapter, we will be talking about the homogeneous differential equation of the first order.

**Homogeneous Functions:**

If f is a function of n number of variables defined on a set S for which (tx_{1}, …, tx_{n}) ∈ S whenever t > 0 and (x_{1}, …, x_{n}) ∈ S and f(tx_{1}, …, tx_{n}) = tk f(x_{1}, …, x_{n}) for all (x_{1}, …, x_{n}) ∈ S and all t > 0. Then, the function is a homogeneous function.

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

Any first-order differential equation, that may be represented as dy = f(x,y) dx

is called a homogeneous differential equation if the function on the R.H.S. is homogeneous in nature, of degree = 0. Hence we can say that for any real number α

f(αx, αy) = α f(x, y) = f(x,y)

Another way of representing of the differential equation can be found by rewriting the homogeneous function on the R.H.S. in the expression of two homogeneous functions M(x,y) and N(x, y) of the same degree –

dy/dx=M(x, y)/N(x, y)

N(x,y) dy = − M(x,y) dx

M(x,y) dx + N(x,y) dy = 0

Another representation is possible, which is only valid for the case of the first order is shown below –

dy/dx=f(x)/y

**Equations Reducible to the Homogeneous Form:**

The first-order differential equations of the form –

here a, b, c, k, l, m are constants. On rewriting the equation –

This is the Homogeneous Form itself if c = m = 0. In order to do that, we introduce new variables in the system X = x – α, Y = y – β

here α and β are constants that we will find from the condition c = m = 0. This method will convert the equation to a homogeneous form which then can be directly solved by the substitution Y=vX. Although we have to back-substitute Y in the equation Y = y – β to get the final solution.

**Highlights:**

Methods of Solving a Homogeneous Differential Equation:

- Introduce a new dependent variable v=yx.
- The differential equation now becomes – x dv/dx + v = f(1)/v

- Solve this differential equation by the variables separation method.
- Replace the statement obtained for v back in y=vx to find the general solution to the differential equation.

### Exercise Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 20 Homogeneous Differential Equation

- In the exercise solutions of RS Aggarwal Class 12 Chapter 20, you will have to check and prove if a given differential equation is a homogeneous or an ordinary differential equation.
- In the questions of the exercise, you will also be required to solve the differential equations by introducing a new variable and using the variable separable technique.
- These questions will help you score full marks from the questions of this chapter in your school exams.
- You will also learn quick tricks to approach the objective type questions that are common in the national level exams.
- By solving the exercise questions of this chapter, you will get an idea about what type of questions are asked in different exams of your level.
- Solving this chapter exercise will help you in a quick and thorough revision drill before your exam.

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