Differential Equations — Previous Year Questions (Class 12 Mathematics)
Differential equations describe relationships between variables and their rates of change. They are essential in modeling real-world phenomena in physics,
TL;DR: Differential equations describe relationships between variables and their rates of change. They are essential in modeling real-world phenomena in phys…
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Differential equations describe relationships between variables and their rates of change. They are essential in modeling real-world phenomena in physics,
Differential Equations — Previous Year Questions with Solutions
Q (2023, 2 marks): Define a differential equation. Distinguish between order and degree of a differential equation.
Answer: Differential equation: An equation involving one or more derivatives of a function.
Example: dy/dx + 2y = 3x
Order: The order of the highest derivative present in the equation.
Example: d²y/dx² + 3dy/dx + 2y = 0 has order 2 (highest derivative is d²y/dx²)
Degree: The power (exponent) of the highest derivative after the equation is cleared of fractions and radicals.
Example: (d²y/dx²)² + 3(dy/dx) + 2y = 0 has degree 2
Difference:
Order tells which derivative is highest.
Degree tells the power of that highest derivative.
Order is always a positive integer.
Degree may not exist if the equation cannot be made polynomial in derivatives.
Q (2022, 2 marks): Solve the differential equation dy/dx = 2x with initial condition y = 1 when x = 0.
Answer: Given: dy/dx = 2x, y(0) = 1
Separating variables:
dy = 2x dx
Integrating both sides:
∫dy = ∫2x dx
y = x² + C
Applying initial condition y = 1 when x = 0:
1 = 0² + C
C = 1
Therefore, the solution is:
y = x² + 1
Q (2023, 2 marks): Solve the differential equation dy/dx = 3y using separation of variables.
Answer: Given: dy/dx = 3y (separable differential equation)
Separating variables:
dy/y = 3 dx
Integrating both sides:
∫(1/y) dy = ∫3 dx
ln|y| = 3x + C
Exponentiating both sides:
|y| = e^(3x + C)
|y| = e^C × e^(3x)
y = Ae^(3x), where A = ±e^C
Therefore, the general solution is:
y = Ae^(3x), where A is an arbitrary constant
Q (2022, 3 marks): Solve the differential equation dy/dx + 2y = 4x. (Linear first-order DE)
Answer: Given: dy/dx + 2y = 4x (linear form: dy/dx + P(x)y = Q(x))
where P(x) = 2, Q(x) = 4x
Step 1: Find integrating factor
I.F. = e^(∫P(x)dx) = e^(∫2 dx) = e^(2x)
Step 2: Multiply the equation by I.F.
e^(2x) × (dy/dx + 2y) = e^(2x) × 4x
e^(2x) dy/dx + 2e^(2x) y = 4xe^(2x)
Step 3: Recognize left side as d/dx[e^(2x) × y]
d/dx[e^(2x) × y] = 4xe^(2x)
Step 4: Integrate both sides
e^(2x) × y = ∫4xe^(2x) dx
Using integration by parts: Let u = 4x, dv = e^(2x) dx
du = 4 dx, v = e^(2x)/2
∫4xe^(2x) dx = 4x × e^(2x)/2 - ∫4 × e^(2x)/2 dx = 2xe^(2x) - 2∫e^(2x) dx
= 2xe^(2x) - e^(2x) + C
e^(2x) × y = 2xe^(2x) - e^(2x) + C
Step 5: Solve for y
y = 2x - 1 + Ce^(-2x)
Q (2023, 2 marks): Verify that y = e^x is a solution to the differential equation d²y/dx² - dy/dx = 0.
Answer: Given: y = e^x
Differential equation: d²y/dx² - dy/dx = 0
Finding derivatives:
dy/dx = e^x
d²y/dx² = e^x
Substituting into the differential equation:
d²y/dx² - dy/dx = e^x - e^x = 0 ✓
Since substituting y = e^x satisfies the equation (LHS = 0 = RHS), y = e^x is indeed a solution.
Q (2021, 3 marks): Solve the differential equation (1 + x²) dy/dx + 2xy = 4x² using integrating factor method.
Answer: Given: (1 + x²) dy/dx + 2xy = 4x²
Step 1: Convert to standard form
Divide by (1 + x²):
dy/dx + (2x/(1 + x²))y = 4x²/(1 + x²)
Step 2: Identify P(x) and Q(x)
P(x) = 2x/(1 + x²), Q(x) = 4x²/(1 + x²)
Step 3: Find integrating factor
I.F. = e^(∫P(x)dx) = e^(∫2x/(1+x²) dx)
= e^(ln(1+x²)) = 1 + x²
Step 4: Multiply by I.F.
(1 + x²) dy/dx + 2xy = 4x²
This is already in the form: d/dx[(1+x²)y] = 4x²
Step 5: Integrate
∫d/dx[(1+x²)y] = ∫4x² dx
(1 + x²)y = (4/3)x³ + C
Step 6: Solve for y
y = (4x³/3 + C)/(1 + x²)
y = 4x³/(3(1 + x²)) + C/(1 + x²)
Frequently Asked Questions
What is the difference between a particular solution and a general solution of a differential equation?
A general solution contains arbitrary constants (parameters) and represents a family of curves. For example, y = x² + C is the general solution of dy/dx = 2x. A particular solution is obtained by assigning specific values to these constants using initial conditions. For example, y = x² + 1 (with C = 1) is a particular solution. The number of arbitrary constants equals the order of the differential equation.
When is a differential equation separable and how do you solve it?
A differential equation is separable if it can be written in the form dy/dx = f(x)g(y), where the variables can be separated. To solve: (1) Separate variables: (1/g(y)) dy = f(x) dx, (2) Integrate both sides: ∫(1/g(y)) dy = ∫f(x) dx, (3) Solve for y if possible. Not all differential equations are separable. If a separable form doesn't exist or is difficult to find, other methods (integrating factor, substitution) must be used.
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