Class 12 Mathematics Notes Chapter 9 (Differential Equations) – Examplar Problems (English) Book

Alright class, let's delve into Chapter 9: Differential Equations from your NCERT Exemplar. This chapter is crucial not just for your board exams but forms a significant part of the mathematics syllabus for various government examinations. Pay close attention to the concepts, definitions, and methods of solving these equations.

Chapter 9: Differential Equations - Detailed Notes

1. Introduction: What are Differential Equations?

• An equation involving derivatives (or differentials) of one or more dependent variables with respect to one or more independent variables is called a differential equation.
• Examples:
  • dy/dx = x + sin x (Involves derivative of y w.r.t. x)
  • (d²y/dx²) + y = 0 (Involves second derivative)
  • ∂z/∂x + ∂z/∂y = x + y (Partial differential equation - usually beyond Class 12 scope, but good to know the distinction)
• We primarily deal with Ordinary Differential Equations (ODEs): Equations containing derivatives of only one dependent variable with respect to only one independent variable. The first two examples above are ODEs.

2. Order and Degree of a Differential Equation

• Order: The order of the highest order derivative appearing in the differential equation (after clearing radicals and fractions in derivatives).
  • Example: d³y/dx³ + x²(d²y/dx²)³ = 0. The highest derivative is d³y/dx³, so the Order = 3.
  • Example: dy/dx + y = 1/(dy/dx). Rewrite as (dy/dx)² + y(dy/dx) = 1. Highest derivative is dy/dx, so Order = 1.
• Degree: The power (positive integer) of the highest order derivative, after the equation has been expressed as a polynomial in derivatives (i.e., free from radicals and fractional powers involving derivatives).
  • Example: d³y/dx³ + x²(d²y/dx²)³ = 0. Highest order derivative is d³y/dx³, its power is 1. So, Degree = 1.
  • Example: (d²y/dx²)³ + (dy/dx)⁴ + y⁵ = 0. Highest order derivative is d²y/dx², its power is 3. So, Degree = 3.
  • Example: √(1 + (dy/dx)²) = d²y/dx². Square both sides: 1 + (dy/dx)² = (d²y/dx²)². Highest derivative is d²y/dx², its power is 2. So, Order = 2, Degree = 2.
  • Important Note: If the equation cannot be expressed as a polynomial in derivatives (e.g., sin(dy/dx) + y = 0), the degree is not defined.

3. Formation of a Differential Equation

• A differential equation can be formed from its general solution (a relation between variables involving arbitrary constants) by eliminating these constants.
• Rule: If the general solution has 'n' arbitrary constants, differentiate the solution 'n' times. Eliminate the 'n' constants using the original equation and the 'n' derived equations. The resulting equation will be a differential equation of order 'n'.
• Example: Form the DE for y = A cos x + B sin x (A, B are arbitrary constants).
  • Two constants (A, B), so we need a 2nd order DE.
  • dy/dx = -A sin x + B cos x
  • d²y/dx² = -A cos x - B sin x = -(A cos x + B sin x) = -y
  • Therefore, the DE is d²y/dx² + y = 0.

4. Solutions of Differential Equations

• General Solution: A solution containing a number of independent arbitrary constants equal to the order of the differential equation.
• Particular Solution: A solution obtained from the general solution by giving particular values to the arbitrary constants. These values are often determined using initial conditions or boundary conditions.
  • Example: For d²y/dx² + y = 0, the general solution is y = A cos x + B sin x. If given y(0) = 1 and y'(0) = 0, we find A=1, B=0. So, y = cos x is a particular solution.
• Singular Solution: A solution that cannot be obtained from the general solution by assigning specific values to the arbitrary constants (less common in standard syllabus).

5. Methods of Solving First Order, First Degree Differential Equations

These are equations of the form dy/dx = F(x, y) or M(x, y) dx + N(x, y) dy = 0.

• (a) Variables Separable:

  • If the equation can be written in the form f(y) dy = g(x) dx.
  • Method: Separate the variables (all y terms with dy, all x terms with dx) and integrate both sides.
  • ∫ f(y) dy = ∫ g(x) dx + C (where C is the constant of integration).
  • Example: Solve dy/dx = (1+y²)/(1+x²).
    • Separate: dy / (1+y²) = dx / (1+x²).
    • Integrate: ∫ dy / (1+y²) = ∫ dx / (1+x²) + C.
    • Solution: tan⁻¹(y) = tan⁻¹(x) + C.

• (b) Homogeneous Differential Equations:

  • A function F(x, y) is homogeneous of degree 'n' if F(λx, λy) = λⁿ F(x, y).
  • A first-order DE dy/dx = F(x, y) is homogeneous if F(x, y) is a homogeneous function of degree zero, OR if it can be written as dy/dx = G(y/x) or dx/dy = H(x/y).
  • Method:
    1. Substitute y = vx. Then dy/dx = v + x(dv/dx).
    2. Substitute these into the original equation. The equation should become variables separable in v and x.
    3. Separate variables: f(v) dv = g(x) dx / x.
    4. Integrate both sides.
    5. Replace v with y/x to get the solution in terms of x and y.
    • (Alternatively, if the form dx/dy = H(x/y) is more convenient, substitute x = vy and proceed similarly).
  • Example: Solve (x² + y²) dx - 2xy dy = 0.
    • Rewrite: dy/dx = (x² + y²) / (2xy) = (1 + (y/x)²) / (2(y/x)). This is homogeneous.
    • Put y = vx, dy/dx = v + x(dv/dx).
    • v + x(dv/dx) = (1 + v²) / (2v).
    • x(dv/dx) = (1 + v²) / (2v) - v = (1 + v² - 2v²) / (2v) = (1 - v²) / (2v).
    • Separate: (2v / (1 - v²)) dv = dx / x.
    • Integrate: ∫ (2v / (1 - v²)) dv = ∫ dx / x + C₁.
    • -ln|1 - v²| = ln|x| + C₁ => ln|x(1 - v²)| = -C₁ = ln|C| (where C = e⁻ᶜ¹).
    • x(1 - v²) = ±C. Let K = ±C.
    • x(1 - (y/x)²) = K => x((x² - y²) / x²) = K => (x² - y²) / x = K.
    • Solution: x² - y² = Kx.

• (c) Linear Differential Equations:

  • An equation of the form dy/dx + P(x) y = Q(x), where P(x) and Q(x) are functions of x only (or constants).
  • Method:
    1. Identify P(x) and Q(x).
    2. Calculate the Integrating Factor (I.F.): I.F. = e^(∫ P(x) dx).
    3. The general solution is given by: y * (I.F.) = ∫ [Q(x) * (I.F.)] dx + C.
  • Alternative Form: dx/dy + P₁(y) x = Q₁(y), where P₁(y) and Q₁(y) are functions of y only (or constants).
    • I.F. = e^(∫ P₁(y) dy).
    • Solution: x * (I.F.) = ∫ [Q₁(y) * (I.F.)] dy + C.
  • Example: Solve dy/dx + (1/x) y = x².
    • This is linear with P(x) = 1/x and Q(x) = x².
    • I.F. = e^(∫ (1/x) dx) = e^(ln|x|) = |x|. Assuming x > 0, I.F. = x.
    • Solution: y * x = ∫ (x² * x) dx + C.
    • xy = ∫ x³ dx + C = x⁴/4 + C.
    • y = x³/4 + C/x.

6. Key Points for Exams

• Be quick in identifying the type of the differential equation (Separable, Homogeneous, Linear). This is the first crucial step.
• Memorize the formulas for Integrating Factors and the structure of the general solutions for Linear DEs.
• Practice integration thoroughly, as it's essential for solving DEs.
• Pay attention to initial conditions to find particular solutions.
• Understand the meaning of order and degree and how to determine them correctly, especially when radicals or fractions are involved.
• Formation of DEs is also important – know the process of differentiation and elimination.

Multiple Choice Questions (MCQs)

1. The order and degree of the differential equation [1 + (dy/dx)²]^(3/2) = d²y/dx² are respectively:
   (a) 2, 2
   (b) 2, 3
   (c) 2, 1
   (d) 1, 2

2. The differential equation representing the family of curves y = A eˣ + B e⁻ˣ, where A and B are arbitrary constants, is:
   (a) d²y/dx² + y = 0
   (b) d²y/dx² - y = 0
   (c) dy/dx + y = 0
   (d) dy/dx - y = 0

3. The general solution of the differential equation dy/dx = e^(x-y) is:
   (a) eʸ = eˣ + C
   (b) e⁻ʸ = e⁻ˣ + C
   (c) eˣ + eʸ = C
   (d) eˣ - eʸ = C

4. The integrating factor (I.F.) of the differential equation x dy/dx - y = 2x² is:
   (a) x
   (b) -1/x
   (c) 1/x
   (d) log|x|

5. The differential equation dy/dx = (x+y)/x is:
   (a) Variables separable
   (b) Homogeneous
   (c) Linear
   (d) None of these

6. The solution of the differential equation dy/dx + y tan x = sec x, given y(0) = 0, is:
   (a) y = sin x
   (b) y = cos x
   (c) y = tan x
   (d) y = sec x

7. The degree of the differential equation sin(dy/dx) + cos(d²y/dx²) = y is:
   (a) 1
   (b) 2
   (c) 3
   (d) Not defined

8. The differential equation formed by eliminating the arbitrary constant 'a' from (x-a)² + y² = a² is:
   (a) x² + y² - 2xy (dy/dx) = 0
   (b) x² - y² - 2xy (dy/dx) = 0
   (c) x² + y² + 2xy (dy/dx) = 0
   (d) x² - y² + 2xy (dy/dx) = 0

9. Which of the following is a homogeneous differential equation?
   (a) dy/dx = x² + y²
   (b) dy/dx = (y² - x²) / (xy)
   (c) dy/dx = sin x + y
   (d) (x² + 1) dy/dx + 2xy = x

10. A particular solution of dy/dx = y cot x satisfying y(π/2) = 2 is:
    (a) y = 2 sin x
    (b) y = 2 cos x
    (c) y = sin(2x)
    (d) y = 2 tan(x/2)

Answer Key for MCQs:

1. (a)
2. (b)
3. (a)
4. (c)
5. (b)
6. (a)
7. (d)
8. (d)
9. (b)
10. (a)

Study these notes carefully, practice solving problems from your Exemplar and other standard books. Focus on identifying the type of equation first, then applying the correct method systematically. Good luck with your preparation!