RD Sharma Solutions

RD Sharma Class 12 Solutions Chapter 22 - Differential Equations (Ex 22.10) Exercise 22.10 - Free PDF

RD Sharma Class 12 Solutions Chapter 22 - Differential Equations (Ex 22.10) Exercise 22.10 - Free PDF

Q: 1. What is the standard method for solving questions in RD Sharma Class 12 Solutions Exercise 22.10?

The questions in Exercise 22.10 are based on linear differential equations. The standard method involves these steps:First, rearrange the given equation into the standard form: dy/dx + Py = Q, where P and Q are functions of x.Next, calculate the Integrating Factor (I.F.) using the formula: I.F. = e∫P dx.Finally, apply the general solution formula: y × (I.F.) = ∫(Q × I.F.) dx + C, where C is the constant of integration.

Q: 4. What is a common mistake to avoid when solving problems from Exercise 22.10?

A very common mistake is forgetting to add the constant of integration, 'C', after performing the integration on the right-hand side of the general solution formula [y × (I.F.) = ∫(Q × I.F.) dx]. Omitting 'C' results in a particular solution, not the required general solution, and will lead to an incorrect final answer.

Q: 5. How does the solution method change for linear differential equations of the form dx/dy + Px = Q?

When the equation is in the form dx/dy + Px = Q, the roles of x and y are interchanged. Here, P and Q are functions of 'y' or constants. The method is analogous:The Integrating Factor is calculated with respect to y: I.F. = e∫P dy.The general solution becomes: x × (I.F.) = ∫(Q × I.F.) dy + C.This highlights the importance of correctly identifying the independent and dependent variables first.

Preparation for Class 12 with Solutions

Free PDF download of RD Sharma Class 12 Solutions Chapter 22 - Differential Equations Exercise 22.10 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 22 - Differential Equations Ex 22.10 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.

Differential Equations

A differential equation may be defined as an equation that consists of the derivative of an unknown equation. The derivatives of the function helps to find the rate of change of a function. A differential equation is an equation that relates the derivatives with the other functions. Differential equations are mostly helpful in the fields of biology, physics, engineering, and many more. Studying the solutions that satisfy the given equations and the properties of solutions is one of the main purposes of the differential equations.

What are the Differential Equations?

A differential equation can be defined as the equation that consists of at least one derivative of an unknown function, either an ordinary or a partial derivative. Suppose, the rate of change of a function y concerning x is inversely proportional to y, we express it as dy/dx = k/y.

A Differential equation can also be defined as an equation that includes the derivative (derivatives) of the dependent variable concerning the independent variable (variables) in calculus. The derivative helps to represent a rate of change, and the differential equation helps us represent a relationship between the changing quantity concerning the change in another quantity. Let y=f(x) be a function where y is a dependent variable, f is an unknown function, x is an independent variable.

Order of Differential Equations

The order of a differential equation helps to find the highest order of the derivative appearing in the given equation. Depending on the order, there are two types of differential equations

First-order differential equation

Second-order differential equation

Degree of Differential Equations

If we can express a differential equation in the terms of a polynomial form, then the integral power of the highest order derivative that appears is called the degree of the differential equation. The power of the highest order derivative given in the equation is the degree of the differential equation. To calculate the degree of the differential equation, we will require a positive integer as the index of each derivative. If a differential equation cannot be expressed in the terms of a polynomial equation having the same highest-order derivative as the leading term, then, the degree of the differential equation is not defined.

FAQs on RD Sharma Class 12 Solutions Chapter 22 - Differential Equations (Ex 22.10) Exercise 22.10 - Free PDF

1. What is the standard method for solving questions in RD Sharma Class 12 Solutions Exercise 22.10?

The questions in Exercise 22.10 are based on linear differential equations. The standard method involves these steps:

First, rearrange the given equation into the standard form: dy/dx + Py = Q, where P and Q are functions of x.
Next, calculate the Integrating Factor (I.F.) using the formula: I.F. = e^{∫P dx}.
Finally, apply the general solution formula: y × (I.F.) = ∫(Q × I.F.) dx + C, where C is the constant of integration.

2. How do you identify the functions P and Q in a linear differential equation?

To identify P and Q, you must first write the equation in the standard form dy/dx + Py = Q. The term 'P' is the coefficient or function multiplied by 'y', and the term 'Q' is the function on the right-hand side of the equation. For example, in the equation dy/dx + (2/x)y = x, P is 2/x and Q is x.

3. Why is the Integrating Factor (I.F.) crucial for solving linear differential equations?

The Integrating Factor (I.F.) is a special function that is used to simplify the differential equation. When you multiply the entire equation in the form dy/dx + Py = Q by the I.F., the left-hand side transforms into the exact derivative of the product y × (I.F.). This strategic conversion, d/dx [y × (I.F.)] = Q × (I.F.), makes the equation directly integrable, allowing you to find the general solution.

4. What is a common mistake to avoid when solving problems from Exercise 22.10?

A very common mistake is forgetting to add the constant of integration, 'C', after performing the integration on the right-hand side of the general solution formula [y × (I.F.) = ∫(Q × I.F.) dx]. Omitting 'C' results in a particular solution, not the required general solution, and will lead to an incorrect final answer.

5. How does the solution method change for linear differential equations of the form dx/dy + Px = Q?

When the equation is in the form dx/dy + Px = Q, the roles of x and y are interchanged. Here, P and Q are functions of 'y' or constants. The method is analogous:

The Integrating Factor is calculated with respect to y: I.F. = e^{∫P dy}.
The general solution becomes: x × (I.F.) = ∫(Q × I.F.) dy + C.

This highlights the importance of correctly identifying the independent and dependent variables first.

6. Can every first-order differential equation be solved using the linear equation method from Ex 22.10?

No, not every first-order differential equation can be solved using this method. This technique is specifically for equations that are linear in the dependent variable and its derivative. Other types, such as homogeneous differential equations, require different methods like the substitution y = vx. For detailed methods on all types, you can refer to the complete RD Sharma Class 12 Maths Solutions Chapter 22.

7. Where can I find reliable, step-by-step solutions for all questions in RD Sharma Class 12?

Vedantu provides accurate and easy-to-understand solutions for all chapters. For a complete guide covering every exercise and concept according to the latest CBSE guidelines, you can access the RD Sharma Class 12 Mathematics Chapter-Wise Solutions on our website.