Ordinary Differential Equations
Tamil Nadu Board · Class 12 · Mathematics
Flashcards for Ordinary Differential Equations — Tamil Nadu Board Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Explore the full setIdentify the order and degree of the differential equation: (d²y/dx²) + 5(dy/dx) + 6y = 0
Answer
Step 1: Find the highest order derivative present → d²y/dx² (second derivative) Step 2: The order is 2 Step 3: Check the power of the highest order derivative → (d²y/dx²)¹, power = 1 Step 4: The equat…
Find the order and degree of: √(1 + (dy/dx)²) = y(d³y/dx³)
Answer
Step 1: The equation contains a fractional power (square root), so it's not in polynomial form Step 2: Square both sides to eliminate the fractional power: 1 + (dy/dx)² = y²(d³y/dx³)² Step 3: Identify…
Why is degree undefined for: sin(dy/dx) + (d²y/dx²) + 3x = 0?
Answer
Answer: Because the highest order derivative (d²y/dx²) is not trapped in a transcendental function (sin, cos, exp, log), BUT the first derivative IS trapped in the sine function. Key Reason: After ide…
Classify the differential equation: (1 + x³)(dy/dx) + 6x²y = 1 + x²
Answer
Step 1: Rewrite in standard form by dividing by (1 + x³): dy/dx + [6x²/(1 + x³)]y = (1 + x²)/(1 + x³) Step 2: Check linearity criteria: - y appears only to first power ✓ - dy/dx appears only to first…
Which of these is nonlinear and why? (a) y'' + 2xy' = sin x (b) y² + y' = √x (c) y' + p(x)y = 0
Answer
Answer: (b) y² + y' = √x is nonlinear Analysis: (a) y'' + 2xy' = sin x - y'' appears to first power ✓ - y' appears to first power ✓ - No products of y and derivatives ✓ - Coefficient 2x is a function…
Form a differential equation by eliminating the constant 'a' from: y² = 4ax
Answer
Step 1: Given family of parabolas: y² = 4ax, where 'a' is arbitrary constant Step 2: Differentiate with respect to x: 2y(dy/dx) = 4a Step 3: Solve for 'a' from differentiation: a = (1/2)y(dy/dx) St…
Eliminate arbitrary constants A and B from: y = A cos x + B sin x
Answer
Step 1: Given family: y = A cos x + B sin x (contains 2 constants) Step 2: First differentiation: dy/dx = -A sin x + B cos x ... (equation 2) Step 3: Second differentiation: d²y/dx² = -A cos x - B s…
Solve using variables separable: (1 + x²)dy/dx = 1 + y²
Answer
Step 1: Recognize this is a separable equation (variables can be separated) Step 2: Rearrange to separate variables: dy/(1 + y²) = dx/(1 + x²) Step 3: Integrate both sides: ∫ dy/(1 + y²) = ∫ dx/(1 +…
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