Order & Degree
Differential Equations
12
JEE Qs
8%
Hard
45
min
Always ensure the differential equation is free from radicals and fractions involving derivatives, and is a polynomial in derivatives before determining the degree; otherwise, the degree is undefined.
🧮 Key Formulas
✅ Key Points for JEE
- 1Order is always well-defined for any differential equation, as long as derivatives exist.
- 2For degree to be defined, the differential equation must be expressible as a polynomial in all its derivatives (y', y'', y'''...).
- 3Before determining the degree, always clear all radicals (square roots, cube roots) and fractional powers involving derivatives by suitable algebraic manipulation (e.g., squaring both sides).
- 4If any derivative is an argument of a transcendental function (e.g., sin(dy/dx), e^(d²y/dx²), log(d³y/dx³)), then the differential equation cannot be expressed as a polynomial in derivatives, and thus its degree is undefined.
- 5The degree is the power of the *highest order derivative* only, after ensuring it's a polynomial in derivatives. Do not confuse it with the highest power of any derivative present.
⚠️ Common Mistakes
- ✕Failing to clear radicals or fractional powers involving derivatives before determining the degree, leading to an incorrect degree value.
- ✕Incorrectly stating the degree when derivatives are inside transcendental functions (e.g., assigning degree 1 to sin(dy/dx) = x+y).
- ✕Confusing the highest power of any derivative with the power of the highest order derivative.
- ✕Mixing up the definitions of order and degree under exam pressure.
📝 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 Part II Ch 9: Differential Equations