📐 CHAPTER 9: DIFFERENTIAL EQUATIONS
Subject: Mathematics | Class: XII | CBSE
📖 INTRODUCTION
Differential Equation
An equation involving:
- Independent variable
- Dependent variable
- Derivatives of dependent variable with respect to independent variable
- Constants
is called a differential equation.
Examples
- dy/dx = x² + 3
- d²y/dx² + 2(dy/dx) + y = 0
- dy/dx + y = eˣ
🔹 Ordinary Differential Equation
Ordinary Differential Equation
An equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.
💡 Note: From any given relationship between the dependent and independent variables, a differential equation can be formed by differentiating it and eliminating arbitrary constants.
📊 ORDER AND DEGREE
🔹 Order of a Differential Equation
Order
The order of a differential equation is the order of the highest order derivative present in the equation.
💡 Note: Order of a differential equation cannot be more than the number of arbitrary constants in the equation.
Example
d²y/dx² + 3(dy/dx) + 2y = 0 → Order = 2
🔹 Degree of a Differential Equation
Degree
The degree of a differential equation is the highest exponent of the highest order derivative, provided the equation is a polynomial equation in derivatives.
💡 Notes:
- Order and degree (if defined) are always positive integers.
- The differential equation must be a polynomial equation in derivatives.
- If the equation is not a polynomial in derivatives, its degree is not defined.
Example
d²y/dx² + 3(dy/dx) + 2y = 0 → Degree = 1
(d²y/dx²)³ + (dy/dx)² + y = 0 → Degree = 3
📝 FORMATION OF A DIFFERENTIAL EQUATION
Steps to Form a Differential Equation
Step 1: Write the given equation and count the number of arbitrary constants.
Step 2: Differentiate the given equation with respect to the dependent variable n times, where n is the number of arbitrary constants.
Step 3: Eliminate all arbitrary constants from the equations formed after differentiation.
Step 4: The equation obtained without arbitrary constants is the required differential equation.
Example
Form the differential equation from y = A sin x + B cos x.
Solution: Differentiating: dy/dx = A cos x - B sin x
Differentiating again: d²y/dx² = -A sin x - B cos x = -y
∴ d²y/dx² + y = 0
📐 SOLUTION OF DIFFERENTIAL EQUATION
Solution
A function of the form y = Φ(x) + C that satisfies the given differential equation is called a solution.
🔹 General Solution
General Solution
The solution containing as many arbitrary constants as the order of the differential equation is called the general solution.
🔹 Particular Solution
Particular Solution
A solution obtained by giving particular values to arbitrary constants in the general solution is called a particular solution.
🔀 VARIABLE SEPARABLE METHOD
Variable Separable Form
If dy/dx = h(x)·k(y), then:
dy/k(y) = h(x) dx
Integrate both sides:
K(y) = H(x) + C
Example
dy/dx = x/y
y dy = x dx
∫y dy = ∫x dx
y²/2 = x²/2 + C
y² - x² = C
📊 HOMOGENEOUS DIFFERENTIAL EQUATIONS
Homogeneous Differential Equation
A differential equation dy/dx = f(x,y)/g(x,y) is homogeneous if f(x,y) and g(x,y) are homogeneous functions of the same degree.
💡 Test: Replace x by λx and y by λy. If F(λx, λy) = F(x,y), the equation is homogeneous.
🔹 Method of Solution
To solve a homogeneous differential equation:
- Put y = vx
- Then dy/dx = v + x(dv/dx)
- Substitute to get a variable separable form
- Solve and replace v = y/x
Example
dy/dx = (x² + y²)/(xy)
Put y = vx: dy/dx = v + x(dv/dx)
v + x(dv/dx) = (x² + v²x²)/(vx²) = (1+v²)/v
x(dv/dx) = (1+v²)/v - v = 1/v
v dv = dx/x
v²/2 = log|x| + C
💡 Note: If the homogeneous equation is of the form dx/dy = F(x,y), put x = vy instead.
📈 LINEAR DIFFERENTIAL EQUATIONS
Linear Differential Equation
General form:
dy/dx + P·y = Q
where P and Q are functions of x or constants.
Alternative form:
dx/dy + P'·x = Q'
where P' and Q' are functions of y or constants.
🔹 Integrating Factor (IF)
IF = e∫P dx
or IF = e∫P' dy
🔹 Solution of Linear Differential Equation
y × IF = ∫(Q × IF) dx + C
or
x × IF = ∫(Q' × IF) dy + C
Example
dy/dx + y = eˣ
Here, P = 1, Q = eˣ
IF = e∫1 dx = eˣ
y·eˣ = ∫eˣ·eˣ dx = ∫e²ˣ dx = e²ˣ/2 + C
y = eˣ/2 + C·e⁻ˣ
📝 MCQ QUESTIONS
Q1. The order of the differential equation d²y/dx² + 3(dy/dx) + 2y = 0 is:
(a) 1 (b) 2 (c) 3 (d) 0
Answer: (b) 2
Q2. The degree of the differential equation (d²y/dx²)³ + (dy/dx)² + y = 0 is:
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) 3
Q3. The solution of dy/dx = x/y is:
(a) y² - x² = C (b) y² + x² = C (c) y = x² + C (d) None
Answer: (a) y² - x² = C
Q4. The integrating factor for dy/dx + y = eˣ is:
(a) eˣ (b) e⁻ˣ (c) e²ˣ (d) 1
Answer: (a) eˣ
Q5. The differential equation formed by eliminating arbitrary constants from y = A sin x is:
(a) dy/dx + y = 0 (b) dy/dx - y = 0 (c) dy/dx = cot x (d) dy/dx = y cot x
Answer: (d) dy/dx = y cot x
Solution: y = A sin x ⇒ dy/dx = A cos x = y cot x
Q6. The order of d³y/dx³ + 2(d²y/dx²) + 3(dy/dx) + y = 0 is:
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) 3
Q7. The degree of (dy/dx)⁴ + 3x²(dy/dx)² + y = 0 is:
(a) 2 (b) 3 (c) 4 (d) 1
Answer: (c) 4
Q8. A differential equation is called homogeneous if:
(a) f(x,y) and g(x,y) are homogeneous of same degree (b) f(x,y) = g(x,y) (c) f(x,y) = 0 (d) None
Answer: (a) f(x,y) and g(x,y) are homogeneous of same degree
Q9. For a homogeneous differential equation, the substitution used is:
(a) y = vx (b) x = vy (c) Both A and B (d) None
Answer: (c) Both A and B
Q10. The integrating factor for dy/dx + 2xy = x is:
(a) eˣ² (b) e⁻ˣ² (c) e²ˣ (d) eˣ
Answer: (a) eˣ²
Solution: IF = e∫2x dx = eˣ²
Q11. The general solution of dy/dx = eˣ is:
(a) y = eˣ + C (b) y = -eˣ + C (c) y = eˣ (d) None
Answer: (a) y = eˣ + C
Q12. The particular solution of a differential equation is obtained by:
(a) Giving particular values to arbitrary constants (b) Differentiating (c) Integrating (d) None
Answer: (a) Giving particular values to arbitrary constants
Q13. The order of a differential equation is:
(a) Highest exponent of derivative (b) Highest order derivative (c) Number of constants (d) None
Answer: (b) Highest order derivative
Q14. The degree of a differential equation is defined only if:
(a) It is polynomial in derivatives (b) It is linear (c) It is homogeneous (d) None
Answer: (a) It is polynomial in derivatives
Q15. The solution of dy/dx = y is:
(a) y = eˣ + C (b) y = Ceˣ (c) y = eˣ (d) y = C + eˣ
Answer: (b) y = Ceˣ
✅ ANSWER KEY - MCQs
| Q.No. | Answer | Q.No. | Answer |
|---|---|---|---|
| 1 | (b) | 9 | (c) |
| 2 | (c) | 10 | (a) |
| 3 | (a) | 11 | (a) |
| 4 | (a) | 12 | (a) |
| 5 | (d) | 13 | (b) |
| 6 | (c) | 14 | (a) |
| 7 | (c) | 15 | (b) |
| 8 | (a) |
📌 KEY FORMULAS SUMMARY
| Concept | Formula / Method |
|---|---|
| Order | Highest order derivative present |
| Degree | Highest exponent of highest order derivative (polynomial form) |
| Variable Separable | dy/k(y) = h(x) dx |
| Homogeneous | Put y = vx (or x = vy) |
| Linear Form | dy/dx + P·y = Q |
| Integrating Factor (IF) | IF = e∫P dx |
| Solution of Linear DE | y·IF = ∫(Q·IF) dx + C |
| Alternative Linear Form | dx/dy + P'·x = Q' |
| IF (Alternative) | IF = e∫P' dy |
| Solution (Alternative) | x·IF = ∫(Q'·IF) dy + C |
🔹 Quick Reference:
- Formation: Differentiate and eliminate constants
- General Solution: Contains n arbitrary constants (n = order)
- Particular Solution: Specific values for constants
- Linear DE: Always has an integrating factor
📚 CBSE | Class XII Mathematics
Chapter 9: Differential Equations | Study Notes