Linear Differential Equations — IF method
Differential Equations
12
JEE Qs
8%
Hard
75
min
Master the systematic approach of identifying the form, calculating IF, and applying the solution formula, paying extreme attention to integration details and variable dependencies.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always reduce the differential equation to one of the two standard linear forms: dy/dx + P(x)y = Q(x) or dx/dy + P(y)x = Q(y).
- 2Ensure the coefficient of dy/dx (or dx/dy) is exactly 1 before identifying P and Q.
- 3Carefully identify P(x) and Q(x) (or P(y) and Q(y)) including their signs.
- 4The integrating factor (IF) is e raised to the power of the integral of P; errors in integrating P will propagate.
- 5Remember to include the constant of integration 'C' in the final solution and use given initial conditions to find its value for a particular solution.
⚠️ Common Mistakes
- ✕Incorrectly identifying P and Q, especially when the equation is not in standard form or has incorrect signs.
- ✕Making integration errors while calculating the Integrating Factor (∫Pdx or ∫Pdy) or the final integral (∫Q*IF dx or ∫Q*IF dy).
- ✕Forgetting to make the coefficient of dy/dx (or dx/dy) unity by dividing the entire equation by its coefficient.
- ✕Confusing the variables: for dy/dx + Py = Q, P and Q must be functions of x (or constants); for dx/dy + Px = Q, P and Q must be functions of y (or constants).
📝 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