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Match each item in List-I with the correct value from List-II. List-I: (P) The degree alpha and order beta of [1 + (dy/dx)²]^(3/2) = k*(d²y/dx²), k != 0. Find alpha + beta. (Q) The order of the ODE for y = tan(pi/4 + a*x)*tan(pi/4 - a*x) + c*e^(b*x + d), where a,b,c,d are arbitrary parameters. (R) lim(x->inf) y(x), where dy/dx = y - y² and y(0) = 2. (S) The order alpha and degree beta of the ODE of all tangent lines to x² = 4y. Find alpha + beta. List-II: (1) 1 (2) 2 (3) 3 (4) 4
- 2
- 3
- 4
- 1
Correct answer: 4
Solution
(P) Square: [1+(y')²]³ = k²*(y'')². Order=2, degree=2. alpha+beta=4. -> (4). (Q) tan(pi/4+ax)tan(pi/4-ax)=1 (product of supplementary tangents). So y=1+c*e^(bx+d)=1+C*e^(bx) with 2 parameters (C and b). Order=2. -> (2). (R) dy/dx=y-y². Equilibria at y=0 (unstable) and y=1 (stable). y(0)=2>1: y decreases toward 1. lim y=1. -> (1). (S) Tangent to x²=4y at (t,t²/4): y=tx/2-t²/4. With slope p=dy/dx=t/2 => t=2p. So y=px-p². First-order (alpha=1), degree 2 (beta=2). alpha+beta=3. -> (3). Matching: P->4, Q->2, R->1, S->3.
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