Variable Separable
Differential Equations
12
JEE Qs
8%
Hard
60
min
Master all integration techniques, as accurate and efficient integration is the most crucial skill for correctly solving variable separable differential equations.
🧮 Key Formulas
✅ Key Points for JEE
- 1Identify differential equations where terms involving 'x' and 'dx' can be completely separated from terms involving 'y' and 'dy'.
- 2Algebraically manipulate the equation to bring all 'y' terms (including dy) to one side and all 'x' terms (including dx) to the other side.
- 3Integrate both sides independently after separation. Remember to add a single arbitrary constant 'C' on only one side (usually with the x-integral).
- 4For particular solutions, substitute the given initial conditions (x0, y0) into the general solution to find the specific value of C.
- 5Be mindful of implicit solutions vs. explicit solutions. Sometimes it's difficult or impossible to express y explicitly as a function of x.
⚠️ Common Mistakes
- ✕Forgetting to add the constant of integration 'C' after integrating, or adding 'C' to both sides redundantly.
- ✕Incorrect algebraic manipulation while attempting to separate variables, especially with expressions like (x+y) or (xy).
- ✕Errors in the integration step itself, requiring strong command over indefinite integration techniques.
- ✕Not using absolute values in logarithms (e.g., integral(1/y dy) = ln|y|), which can lead to domain issues or incorrect solutions.
📝 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