Linear Differential Equations (IF Method)
Differential Equations
62
JEE Qs
22%
Hard
75
min
Master the art of identifying the correct linear form and meticulously execute the integration steps for both the Integrating Factor and the final solution.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always reduce the given differential equation to one of the standard linear forms (dy/dx + P(x)y = Q(x) or dx/dy + P(y)x = Q(y)) before proceeding.
- 2Carefully identify P(x)/P(y) and Q(x)/Q(y), including their signs, from the standard form.
- 3The integrating factor (IF) calculation often involves a straightforward integral, but accuracy is paramount as it impacts the entire solution.
- 4The final step of integrating (Q * IF) is usually the most involved and requires strong proficiency in various integration techniques.
- 5Always remember to add the constant of integration 'C' to the final general solution after the last integration step.
⚠️ Common Mistakes
- ✕Incorrectly identifying P(x)/P(y) or Q(x)/Q(y), leading to errors in the Integrating Factor and solution.
- ✕Making calculation errors while integrating P(x) or P(y) to find the Integrating Factor, or during the final integration of (Q * IF).
- ✕Forgetting to include the constant of integration 'C' in the final general solution.
- ✕Trying to apply the linear differential equation method to non-linear equations without making an appropriate substitution (e.g., Bernoulli's form).
- ✕Confusing the variable of integration; if the form is dy/dx, the IF and final integration are with respect to x, and vice-versa for dx/dy.
📝 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