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Class 12 Science (PCM)

Mathematics Part 2

Differential Equations

Differential Equations

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Applications: Continuous Growth and Decay
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Differential Equation Notation
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Eliminating Arbitrary Constants
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Formation of Differential Equations
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Integrating Factor for dx/dy + P₁x = Q₁
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Ordinary Differential Equations Only
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Particular Solution from Initial Condition
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Polynomial Condition for Degree
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Number of Arbitrary Constants in General Solution
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Verification of Solution of Differential Equation
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Integrating Factor Method
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Degree of Differential Equation
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Linear DE Standard Form Conversion
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Variable Separable Method
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Medium
Choosing the Method of Solving Differential Equations
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Differential Equations
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Equation of Curve from Slope Condition
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First Order First Degree Differential Equations
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General and Particular Solutions
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Homogeneous DE Substitution y = vx
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Homogeneous Differential Equations
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Homogeneous Function Recognition
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Integrating Factor for dy/dx + Py = Q
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Linear Differential Equations
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Order and Degree Together
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Order of Differential Equation
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Learning Objectives

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Learning Objectives

  • Understand the definition of differential equations.
  • Identify the order and degree of differential equations.
  • Distinguish between general and particular solutions of differential equations.
  • Apply methods to solve first-order differential equations.
  • Recognize the importance of differential equations in various scientific fields.
  • Solve homogeneous differential equations using appropriate substitutions.
  • Utilize the method of separation of variables for solving differential equations.
  • Analyze the applications of differential equations in real-world scenarios.

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Revision Notes & Summary

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Chapter 9: Differential Equations

9.1 Introduction

  • Definition: A differential equation is an equation involving derivatives of a dependent variable with respect to independent variable(s).
  • Importance: Differential equations arise in various fields such as Physics, Chemistry, Biology, and Economics.
  • Objective: This chapter covers basic concepts, general and particular solutions, formation, methods to solve first order - first degree differential equations, and applications.

9.2 Basic Concepts

9.2.1 Order of a Differential Equation

  • Definition: The order of a differential equation is the order of the highest derivative present.
  • Examples:
    • Equation (6): Order 1
    • Equation (9): Order 2
    • Equation (10): Order 1
    • Equation (11): Order not defined

9.2.2 Degree of a Differential Equation

  • Definition: The degree is defined if the differential equation is a polynomial in its derivatives.
  • Examples:
    • Equation (9): Degree 1
    • Equation (10): Degree 2
    • Equation (11): Degree not defined

9.3 General and Particular Solutions of a Differential Equation

  • General Solution: Contains arbitrary constants equal to the order of the differential equation.
  • Particular Solution: Free from arbitrary constants.
  • Example: For the equation dy/dx = g(x), the solution curve is y = Φ(x).

9.4 Methods of Solving Differential Equations

  • Variable Separable Method: Used when variables can be separated completely.
  • Homogeneous Differential Equations: Can be expressed in the form P(y) = Q(x).

9.5 Applications of Differential Equations

  • Example: The volume of a spherical balloon changes at a constant rate. If the radius is initially 3 units and after 3 seconds it is 6 units, find the radius after t seconds.

Exercises

  1. Find a particular solution for the differential equation dy/dx = y tan x; y = 1 when x = 0.
  2. Verify that the function y = c₁ e^(ax) cos(bx) is a solution of the differential equation.
  3. Find the general solution of the differential equation dy/dx + x²y² + y + 1 = 0.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

Approximately 14 years

B.

Approximately 15 years

C.

Approximately 16 years

D.

Approximately 17 years
Correct Answer: B

Solution:

The differential equation for continuous growth is given by dPdt=0.05P\frac{dP}{dt} = 0.05P. Solving this, we find that the time required for the principal to double is t=ln⁡(2)0.05t = \frac{\ln(2)}{0.05}, which is approximately 13.86 years, rounded to 15 years.

Chapter Concept:

Applications: Continuous Growth and Decay

A.

dy2−y=(x+1)dx\frac{dy}{2 - y} = (x + 1) dx

B.

2−ydy=x+1\frac{2 - y}{dy} = x + 1

C.

dy=(2−y)(x+1)dxdy = (2 - y)(x + 1) dx

D.

dydx=(2−y)(x+1)\frac{dy}{dx} = (2 - y)(x + 1)
Correct Answer: A

Solution:

To separate the variables, rewrite the equation as dy2−y=(x+1)dx\frac{dy}{2 - y} = (x + 1) dx.

Chapter Concept:

Variable Separable Method

A.

y=vxy = vx

B.

x=vyx = vy

C.

y=x+vy = x + v

D.

x=y+vx = y + v
Correct Answer: A

Solution:

The substitution y=vxy = vx is used to transform the equation into a separable form, allowing for easier integration.

Chapter Concept:

Homogeneous DE Substitution y = vx

A.

d3ydx3+2(d2ydx2)2−dydx+y=0\frac{d^3 y}{dx^3} + 2 \left(\frac{d^2 y}{dx^2}\right)^2 - \frac{dy}{dx} + y = 0

B.

(dydx)2+dydx−sin⁡2y=0\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - \sin^2 y = 0

C.

dydx+sin⁡(dydx)=0\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0

D.

d2ydx2+y=0\frac{d^2 y}{dx^2} + y = 0
Correct Answer: C

Solution:

The degree of a differential equation is defined only if the equation is a polynomial in its derivatives. The equation dydx+sin⁡(dydx)=0\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0 is not a polynomial in dydx\frac{dy}{dx}, hence its degree is not defined.

Chapter Concept:

Order and Degree Together

A.

y=vxy = vx

B.

x=vyx = vy

C.

y=x+vy = x + v

D.

x=y+vx = y + v
Correct Answer: A

Solution:

For a homogeneous differential equation of the form dydx=x2+y22xy\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}, the substitution y=vxy = vx simplifies the equation to a function of vv alone, allowing separation of variables.

Chapter Concept:

Equation of Curve from Slope Condition

A.

(dydx)2+dydx−sin⁡2y=0\left( \frac{dy}{dx} \right)^2 + \frac{dy}{dx} - \sin^2 y = 0

B.

dydx+sin⁡(dydx)=0\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0

C.

d3ydx3+2(d2ydx2)2−dydx+y=0\frac{d^3 y}{dx^3} + 2 \left(\frac{d^2 y}{dx^2}\right)^2 - \frac{dy}{dx} + y = 0

D.

d2ydx2+(dydx)2+2y=0\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 2y = 0
Correct Answer: A

Solution:

The degree of a differential equation is defined only if the equation is a polynomial in its derivatives. Option (a) is a polynomial in dydx\frac{dy}{dx}, hence its degree is defined. Option (b) involves a trigonometric function of a derivative, making its degree undefined. Option (c) is a polynomial in its derivatives, so its degree is defined. Option (d) is also a polynomial in its derivatives, so its degree is defined.

Chapter Concept:

Polynomial Condition for Degree

A.

dydx=yx\frac{dy}{dx} = \frac{y}{x}

B.

dydx=x2+y2\frac{dy}{dx} = x^2 + y^2

C.

dydx=sin⁡(x)+cos⁡(y)\frac{dy}{dx} = \sin(x) + \cos(y)

D.

dydx=xy+y\frac{dy}{dx} = \frac{x}{y} + y
Correct Answer: A

Solution:

The equation dydx=yx\frac{dy}{dx} = \frac{y}{x} can be separated as dyy=dxx\frac{dy}{y} = \frac{dx}{x}, making it a variable separable differential equation.

Chapter Concept:

Variable Separable Method

A.

Order: 1, Degree: 1

B.

Order: 3, Degree: 1

C.

Order: 2, Degree: 2

D.

Order: 3, Degree: 2
Correct Answer: B

Solution:

The order of a differential equation is the order of the highest derivative present. In this equation, the highest derivative is d3ydx3\frac{d^3 y}{dx^3}, so the order is 3. The degree is the highest power of the highest order derivative, which is 1.

Chapter Concept:

Differential Equations

A.

P1=tan⁡y,Q1=y2P_1 = \tan y, Q_1 = y^2

B.

P1=y2,Q1=tan⁡yP_1 = y^2, Q_1 = \tan y

C.

P1=cot⁡y,Q1=y2P_1 = \cot y, Q_1 = y^2

D.

P1=y2,Q1=cot⁡yP_1 = y^2, Q_1 = \cot y
Correct Answer: A

Solution:

The differential equation is already in the form dxdy+P1x=Q1\frac{dx}{dy} + P_1x = Q_1 with P1=tan⁡yP_1 = \tan y and Q1=y2Q_1 = y^2.

Chapter Concept:

Linear DE Standard Form Conversion

A.

Order is 2, degree is 1

B.

Order is 1, degree is 2

C.

Order is 2, degree is 2

D.

Order is 1, degree is 1
Correct Answer: A

Solution:

The differential equation d2ydx2+3dydx+2y=0\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0 has the highest derivative of second order, making its order 2. It is a polynomial in its derivatives, and the highest power of the highest derivative is 1, making its degree 1.

Chapter Concept:

Ordinary Differential Equations Only

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True or False

Correct Answer: True

Solution:

The standard form of a first-order linear differential equation is dydx+Py=Q\frac{dy}{dx} + Py = Q, where PP and QQ are constants or functions of xx. This is consistent with the definition provided in the text.

Chapter Concept :

Linear Differential Equations

Correct Answer: True

Solution:

Verification of a solution involves differentiating the proposed function as required, substituting yy and its derivatives into the differential equation, and ensuring that the left-hand side equals the right-hand side.

Chapter Concept :

Verification of Solution of Differential Equation

Correct Answer: True

Solution:

A general solution includes arbitrary constants which allow it to represent a family of curves, as opposed to a particular solution which is derived by assigning specific values to these constants.

Chapter Concept :

General and Particular Solutions

Correct Answer: False

Solution:

The degree of a differential equation is defined only when the equation is a polynomial in its derivatives.

Chapter Concept :

Degree of Differential Equation

Correct Answer: False

Solution:

An ordinary differential equation involves derivatives with respect to only one independent variable. Differential equations involving derivatives with respect to more than one independent variable are called partial differential equations.

Chapter Concept :

Differential Equations

Correct Answer: True

Solution:

In differential equations, y′y' is commonly used to represent the first derivative of the function yy with respect to the variable xx.

Chapter Concept :

Differential Equation Notation

Correct Answer: True

Solution:

A differential equation is considered variable separable if it can be rearranged to separate the variables, allowing integration on both sides.

Chapter Concept :

Variable Separable Method

Correct Answer: False

Solution:

The degree of a differential equation is defined only if the equation is a polynomial in its derivatives. If it is not a polynomial, the degree is not defined.

Chapter Concept :

Order and Degree Together

Correct Answer: False

Solution:

An ordinary differential equation involves derivatives with respect to only one independent variable. Differential equations involving derivatives with respect to more than one independent variable are called partial differential equations.

Chapter Concept :

Ordinary Differential Equations Only

Correct Answer: True

Solution:

The standard form of a first-order linear differential equation is dydx+Py=Q\frac{dy}{dx} + Py = Q, where PP and QQ can be constants or functions of the independent variable xx.

Chapter Concept :

Linear Differential Equations

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