Class 12 RD Sharma Solutions Chapter 22 Differential Equations Exercise 22.1 | Set 2 (original) (raw)

Last Updated : 30 Aug, 2024

Chapter 22 of RD Sharma's Class 12 mathematics textbook introduces students to the fundamental concepts of differential equations, a crucial area of mathematics with wide-ranging applications in physics, engineering, economics, and other sciences. Exercise 22.1 | Set 2 likely serves as a foundational set of problems, focusing on the basics of differential equations. Students are expected to identify and classify differential equations, understand their order and degree, and learn to form differential equations from given relations or conditions.

Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Question 14-26)

Question 14. \sqrt{1-y^2}dx+\sqrt{1-x^2}dy=0

**Solution:

We have,

\sqrt{1-y^2}dx+\sqrt{1-x^2}dy=0

\frac{dy}{dx}+ \sqrt{\frac{1-y^2}{1-x^2}}=0

**Order of function:

As the highest order of derivative of function is 1 (i.e., dy/dx)

So, the order of the derivative is equal to 1.

**Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of dy/dx is 1)

So, the degree of function is 1.

**Linear or Non-linear:

The given equation is non-linear.

Question 15. \frac{d^2y}{dx^2}=(\frac{dy}{dx})^\frac{2}{3}

**Solution:

We have,

\frac{d^2y}{dx^2}=(\frac{dy}{dx})^\frac{2}{3}

On cubing both side, we have

(\frac{d^2y}{dx^2})^3=(\frac{dy}{dx})^2

**Order of function:

The Highest order of derivative of function is 2. (i.e., \frac{d^2y}{dx^2} )

So, the order of the derivative is equal to 2.

**Degree of function:

As the power of the highest order derivative of the function is 3 (i.e., power of \frac{d^2y}{dx^2} is 3)

So, the degree of function is 3.

**Linear or Non-linear:

The given equation is non-linear.

Question 16. 2\frac{d^2y}{dx^2}+3*\sqrt{1-(\frac{dy}{dx})^2-y}=0

**Solution:

We have,

2\frac{d^2y}{dx^2}+3*\sqrt{1-(\frac{dy}{dx})^2-y}

2\frac{d^2y}{dx^2}=-3*\sqrt{1-(\frac{dy}{dx})^2-y}

Squaring both sides, we have

4(\frac{d^2y}{dx^2})^2=9({1-(\frac{dy}{dx})^2-y})

4(\frac{d^2y}{dx^2})^2+9(\frac{dy}{dx})^2+9y-9=0

**Order of function:

As the highest order of derivative of function is 2. (i.e., \frac{d^2y}{dx^2} )

So, the order of the function is equal to 2.

**Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of \frac{d^2y}{dx^2} is 2)

So, the Degree of the function is equal to 2.

**Linear or Non-linear:

The given equation is non-linear.

Question 17. 5\frac{d^2y}{dx^2}=[1+(\frac{dy}{dx})^2]^\frac{3}{2}

**Solution:

We have,

5\frac{d^2y}{dx^2}=[1+(\frac{dy}{dx})^2]^\frac{3}{2}

One squaring both side, we have

[5\frac{d^2y}{dx^2}]^2=[1+(\frac{dy}{dx})^2]^3

25(\frac{d^2y}{dx^2})^2=[1+(\frac{dy}{dx})^6+3(\frac{dy}{dx})^2+3(\frac{dy}{dx})^6

**Order of function:

As the highest order of derivative of the function is 2 (i.e., \frac{d^2y}{dx^2} )

So, Order of the function is equal to 2.

**Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of \frac{d^2y}{dx^2} is 2)

So, the Degree of the function is equal to 2.

**Linear or Non-linear:

The given equation is non-linear.

Question 18. y=x\frac{dy}{dx}+a\sqrt{1+(\frac{dy}{dx})^2}

**Solution:

We have,

y=x\frac{dy}{dx}+a\sqrt{1+(\frac{dy}{dx})^2}

(y-x\frac{dy}{dx})=-a\sqrt{1+(\frac{dy}{dx})^2}

On squaring both sides, we get

(y-x\frac{dy}{dx})^2=a(1+(\frac{dy}{dx})^2

x^2(\frac{dy}{dx})^2+y^2-2xy\frac{dy}{dx}=a[1-(\frac{dy}{dx})^2]

x^2(\frac{dy}{dx})^2+y^2-2xy\frac{dy}{dx}-a+(\frac{dy}{dx})^2=0

**Order of function:

As the highest order of derivative of the function is 1,

So, the Order of the function is equal to 1.

**Degree of function:

As the power of the highest order derivative of the function is 2.(i.e., power of dy/dx is 2)

So, the Degree of the function is equal to 2.

**Linear or Non-linear:

The given equation is non-linear.

Question 19. y=px+\sqrt{a^2p^2+b^2} , where p = dy/dx

**Solution:

We have

y=px+\sqrt{a^2p^2+b^2} , where p = dy/dx

(y-px)^2={a^2p^2+b^2}

y^2-2pxy+p^2x^2=a^2p^2+b^2

(x^2-a^2)(\frac{dy}{dx})^2-2xy(\frac{dy}{dx})+(y^2-b^2)=0

**Order of function:

As the highest order of derivative of function is 1

So, the Order of the function is equal to 1.

**Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of dy/dx is 2)

So, the Degree of the function is equal to 2.

**Linear or Non-linear:

The given equation is non-linear.

Question 20: dy/dx + ey = 0

**Solution:

We have,

dy/dx + ey = 0

**Order of function:

As the highest order of derivative of the function is 1

So, the Order of the function is equal to 1.

**Degree of function:

As the power of the highest order derivative of the function is 1(i.e., power of dy/dx is 1)

So, the Degree of the function is equal to 1.

**Linear or Non-linear:

The given equation is non-linear.

Question 21. (\frac{d^2y}{dx^2})^2+(\frac{dy}{dx})=xsin(\frac{d^2y}{dx^2})

**Solution:

We have,

(\frac{d^2y}{dx^2})^2+(\frac{dy}{dx})=xsin(\frac{d^2y}{dx^2})

**Order of function:

As the highest order of derivative of the function is 2

So, the order of the derivative is equal to 2.

**Degree of function:

sin(\frac{d^2y}{dx^2}) is not a polynomial function. So degree can not be defined.

So, the degree of function is not defined.

**Linear or Non-linear:

The given equation is non-linear.

Question 22. (y'')2 + (y')3 + siny = 0

**Solution:

We have,

(y'')2 + (y')3 + siny = 0

Where

y''=\frac{d^2y}{dx^2}

**Order of function:

The highest order of derivative of the function is 2. (i.e., y'')

So, the order of the derivative is equal to 2.

**Degree of function

As the power of the highest order derivative of the function is 2 (i.e., power of y'' is 2)

So, the degree of function is 2.

**Linear or Non-linear:

The given equation is non-linear.

Question 23.\frac{d^2y}{dx^2}+5x(\frac{dy}{dx})^2-6y=logx

**Solution:

We have,

\frac{d^2y}{dx^2}+5x(\frac{dy}{dx})^2-6y=logx

**Order of function:

As the highest order of derivative of the function is 2.

So, the order of the function is equal to 2.

**Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of \frac{d^2y}{dx^2} is 1)

So, the Degree of the function is equal to 1.

**Linear or Non-linear:

The given equation is non-linear.

Question 24. \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

**Solution:

We have,

\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

**Order of function:

As the highest order of derivative of the function is 3

So, the Order of the function is equal to 3.

**Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of\frac{d^3y}{dx^3} is 1)

So, the Degree of the function is equal to 1.

**Linear or Non-linear:

The given equation is linear.

Question 25. \frac{d^2y}{dx^2}+3(\frac{dy}{dx})^2 =x^2log(\frac{d^2y}{dx^2})

**Solution:

We have,

\frac{d^2y}{dx^2}+3(\frac{dy}{dx})^2 =x^2log(\frac{d^2y}{dx^2})

**Order of function:

As the highest order of derivative of the function is 2.

So, the order of the function is equal to 2.

**The degree of function:

log(\frac{d^2y}{dx^2}) is not a polynomial function. So degree can not be defined.

So, the degree of function is not defined.

**Linear or Non-linear:

The given equation is non-linear.

Question 26. (\frac{dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sinx

**Solution:

We have,

(\frac{dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sinx

\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+ysiny=0

**Order of function:

As the highest order of derivative of the function is 1

So, the Order of the function is equal to 1.

**Degree of function:

As the power of the highest order derivative of the function is 3(i.e., power of dy/dx is 3)

So, the Degree of the function is equal to 3.

**Linear or Non-linear:

The given equation is non-linear.

Summary

This set of practice questions focuses on first-order differential equations, covering various types including separable equations, homogeneous equations, and linear equations. Students will practice identifying the type of differential equation, solving using appropriate methods, and verifying solutions. These questions aim to reinforce understanding of key concepts in differential equations and their applications.

Practice Questions

1. Solve the differential equation: dy/dx = (x + y + 1)/(x - y + 1)

2. Find the general solution of the differential equation: (x^2 + y^2)dx - 2xydy = 0

3. Solve: dy/dx = (x^2 + y^2)/(2xy)

4. Find the particular solution of the differential equation: dy/dx + y/x = x^2, given that y = 1 when x = 1

5. Solve the initial value problem: dy/dx = y/x + x/y, y(1) = 1

6. Find the orthogonal trajectories of the family of curves: y = cx^2

7. Solve the differential equation: (x + y)dx - (x - y)dy = 0

8. Find the general solution of: dy/dx = (x^2 + y^2)^(1/2)

9. Solve: dy/dx = (x + y + 2)/(x + y - 2)

10. Find the particular solution of: dy/dx + (2x/y) = 0, given that y = 2 when x = 0

Summary

This exercise set aims to build a strong foundation in recognizing and working with different types of differential equations, which is essential for solving more complex problems in later sections. By working through these problems, students learn to distinguish differential equations from other types of equations, determine their order and degree, and verify solutions. They also begin to understand the concepts of general and particular solutions, as well as the distinction between linear and non-linear differential equations. This groundwork is crucial for tackling more advanced topics in subsequent sections, such as methods for solving first-order and second-order differential equations. By mastering the concepts presented in Exercise 22.1 | Set 2, students will be well-equipped to explore the more complex aspects of differential equations and their real-world applications in later exercises and chapters.