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Match the differential equations in List-I with their general solutions in List-II. List-I: (P) y - x*(dy/dx) = y² + dy/dx (Q) (2x - 10*y³)*dy + y*dx = 0 (R) sec²(y)*dy + tan(y)*dx = dx (S) sin(y)*(dy/dx) = cos(y)*(1 - x*cos(y)) List-II: (1) x*y² = 2*y⁵ + c (2) sec(y) = x + 1 + c*e^x (3) (x+1)*(1-y) = c*y (4) tan(y) = 1 + c*e^(-x) (5) x²*y + 2*y³ + c = 0 Choose the correct matching.

1. P->3; Q->1; R->2; S->4
2. P->3; Q->1; R->4; S->2
3. P->1; Q->2; R->3; S->4
4. P->3; Q->2; R->4; S->1

Correct answer: P->3; Q->1; R->4; S->2

Solution

P: y - x*(dy/dx) = y² + dy/dx -> y - y² = (x+1)*(dy/dx) -> (dy)/(y(1-y)) = dx/(x+1). Partial fractions: (1/y + 1/(1-y))dy = dx/(x+1). Integrating: ln|y| - ln|1-y| = ln|x+1| + const -> y/(1-y) = A*(x+1) -> (x+1)*(1-y) = c*y. Matches (3). Q: (2x-10y³)dy + y*dx = 0 -> y*dx = (10y³ - 2x)*dy -> dx/dy = (10y³ - 2x)/y = 10y² - 2x/y. This is linear: dx/dy + (2/y)x = 10y². Integrating factor = y². (y²*x)' = 10y⁴. x*y² = 2y⁵ + c. Matches (1). R: sec²(y)*dy + tan(y)*dx = dx -> sec²(y)*dy = (1 - tan(y))*dx. Let t = tan(y): dt = sec²(y)*dy. dt/dx = 1 - t -> dt/dx + t = 1. Linear ODE, IF = e^x. (t*e^x)' = e^x. t*e^x = e^x + c. tan(y) = 1 + c*e^(-x). Matches (4). S: sin(y)*(dy/dx) = cos(y)*(1 - x*cos(y)). Let u = sec(y), du/dx = -sec(y)*tan(y)*... wait: du/dx = d(sec y)/dx = sec(y)*tan(y)*(dy/dx). From the equation: sin(y)*(dy/dx) = cos(y) - x*cos²(y). Divide both sides by cos²(y): tan(y)*sec(y)*(dy/dx) = sec(y) - x. So du/dx = u - x -> du/dx - u = -x. IF = e^(-x). (u*e^(-x))' = -x*e^(-x). Integrating: u*e^(-x) = x*e^(-x) + e^(-x) + c -> u = x + 1 + c*e^x -> sec(y) = x + 1 + c*e^x. Matches (2). Final: P->3, Q->1, R->4, S->2.

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