Class 12 RS Aggarwal Chapter-18 Differential Equations and Their Formation Solutions - Free PDF Download
FAQs on RS Aggarwal Class 12 Solutions Chapter-18 Differential Equations and Their Formation
1. How do the RS Aggarwal Class 12 Solutions for Chapter 18 help with board exam preparation?
The RS Aggarwal solutions for Chapter 18, Differential Equations, provide a structured approach to mastering the topic for the 2025-26 board exams. They offer detailed, step-by-step methods for each problem, which helps in understanding the core logic behind forming and solving different types of equations. By practising with these solutions, you can strengthen your grasp on key concepts like finding the order and degree, using the variable separable method, solving homogeneous equations, and applying the integrating factor for linear equations, ensuring you can tackle any question in the exam with confidence.
2. What is the correct method to determine the order and degree of a differential equation as per Chapter 18?
To find the order and degree of a differential equation, follow these steps as per the CBSE syllabus:
Order: The order is the highest order derivative present in the equation. For example, in d²y/dx² + 5(dy/dx) + 6y = 0, the highest derivative is the second order, so the order is 2.
Degree: The degree is the highest power of the highest order derivative, but only after the equation has been cleared of radicals and fractions in its derivatives. If the equation cannot be expressed as a polynomial in its derivatives (e.g., sin(dy/dx)), the degree is not defined.
It is essential to first identify the highest derivative to find the order, and then examine its power for the degree.
3. What are the key steps for forming a differential equation from a given family of curves?
Forming a differential equation from a given general solution (family of curves) involves eliminating the arbitrary constants. The process is as follows:
Step 1: Count the number of independent arbitrary constants in the given equation. Let this number be 'n'.
Step 2: Differentiate the equation 'n' times successively to get 'n' additional equations.
Step 3: Use all the equations (the original one and the 'n' differentiated ones) to eliminate the 'n' arbitrary constants.
Step 4: The resulting equation, which is free from arbitrary constants, is the required differential equation. Its order will be 'n'.
4. How do you distinguish between a general solution and a particular solution in differential equations?
The main difference lies in the presence of arbitrary constants. A general solution is a solution that contains a number of independent arbitrary constants equal to the order of the differential equation. It represents a family of curves. In contrast, a particular solution is derived from the general solution by assigning specific values to these arbitrary constants, usually based on given initial conditions. A particular solution is free of constants and represents a single, specific curve from that family.
5. Why is it crucial to identify the correct type of differential equation (e.g., variable separable, homogeneous, linear) before solving it?
Identifying the type of a first-order, first-degree differential equation is a critical first step because the solution method is entirely dependent on its form. Each type has a unique algorithm for solving it:
Variable Separable: Requires rearranging the equation so that all terms with 'y' are on one side with dy, and all terms with 'x' are on the other with dx, before integrating.
Homogeneous: Requires the substitution y = vx to transform it into a variable separable form.
Linear (dy/dx + Py = Q): Requires calculating an Integrating Factor (I.F.) to find the solution.
Applying the wrong method will lead to an incorrect solution or a dead end. Correct identification is the foundation of solving differential equations effectively.
6. What is the step-by-step process for solving a linear differential equation of the form dy/dx + Py = Q?
To solve a linear differential equation of the form dy/dx + Py = Q, where P and Q are functions of x, follow these steps:
Step 1: Identify the functions P and Q from the given equation.
Step 2: Calculate the Integrating Factor (I.F.) using the formula: I.F. = e∫P dx.
Step 3: Write down the solution using the standard formula: y × (I.F.) = ∫(Q × I.F.) dx + C.
Step 4: Integrate the right-hand side and simplify to find the general solution of the differential equation. 'C' is the constant of integration.
7. Beyond exams, where are the principles of forming differential equations from Chapter 18 applied in real-world fields?
The principles of forming and solving differential equations are fundamental to modelling real-world phenomena where the rate of change is involved. Key applications include:
Physics: Modelling radioactive decay, Newton's law of cooling, and the motion of objects under various forces.
Biology: Describing population growth (logistic and exponential models) and the spread of diseases.
Engineering: Analysing electrical circuits (LCR circuits) and mechanical systems.
Finance: Modelling compound interest and investment growth over time.
Understanding how to form these equations allows scientists and engineers to predict the future behaviour of a system based on its current state and rate of change.
