Bernoulli Differential Equations
Differential Equations
12
JEE Qs
8%
Hard
60
min
Master the transformation process from the Bernoulli form to a Linear Differential Equation using the correct substitution; this step is critical and often a source of calculation errors.
🧮 Key Formulas
✅ Key Points for JEE
- 1Identify the standard Bernoulli form: dy/dx + P(x)y = Q(x)y^n (or dx/dy + P(y)x = Q(y)x^n). This is crucial for recognizing when to apply the Bernoulli method.
- 2The core strategy involves a specific substitution, v = y^(1-n), which transforms the Bernoulli equation into a standard First Order Linear Differential Equation.
- 3Carefully derive the new differential equation in terms of v and x (or x and y for the other form), ensuring all terms are correctly transformed.
- 4After solving the resulting linear DE for v, always perform the back-substitution (y = v^(1/(1-n))) to express the final solution in terms of y (or x).
- 5Pay close attention to algebraic manipulations and integration steps, as errors here are common.
⚠️ Common Mistakes
- ✕Incorrectly identifying P(x), Q(x), or 'n' from the given equation, leading to an erroneous substitution or transformation.
- ✕Errors in differentiating the substitution (e.g., dv/dx for v = y^(1-n)) or algebraic mistakes while rearranging terms after substitution.
- ✕Miscalculating the Integrating Factor (IF) for the transformed linear differential equation or making errors during the final integration.
- ✕Forgetting to back-substitute v = y^(1-n) (or x = v^(1/(1-n))) at the end, leaving the solution in terms of v instead of the original variable.
📝 Practice Questions
See allQ12.Let f : R →R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y ∈R. If f ′(0) = 4a and f satisfies f ′′(x) −3af ′(x) −f(x) = 0, a > 0, then the area of the region R = {(x, y) ∣0 ≤y ≤f(ax), 0 ≤x ≤2} is: (1) e2 −1 (2) e2 + 1 (3) e4 + 1 (4) e4 −1
Q24.Let y = f(x) be the solution of the differential equation dydx + x2−1xy = √1−x2x6+4x f(0) = 0. If 6 ∫1/2−1/2 f(x)dx = 2π −α then α2 is equal to _______ .
Q2. Let x = x(y) be the solution of the differential equation y2 dx + (x −1y )dy (1) 1 2 + e (2) 3 + e (3) 3 −e (4) 32 + e
Q9. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f ′(y) + f ′(x)f(y) for all x, y ∈R. Then ∑100n=1 loge f(n) is equal to : (1) 2525 (2) 5220 (3) 2384 (4) 2406
Q16.If x = f(y) is the solution of the differential equation (1 + y2) + (x −2etan−1 y) dydx is equal to : f(0) = 1, then f ( √31 ) (1) eπ/12 (2) eπ/4 (3) eπ/3 (4) eπ/6
Q6. Let a curve y = f(x) pass through the points (0, 5) and (loge 2, k). If the curve satisfies the differential equation 2(3 + y)e2xdx −(7 + e2x)dy = 0, then k is equal to (1) 4 (2) 32 (3) 8 (4) 16
NCERT Chapters
- Class 12 Mathematics Ch 9: Differential Equations