If the solution of the differential equation (2x + 3y − 2) dx + (4x + 6y − 7) dy = 0, y(0) = 3, is αx + βy + 3 log_e|2x + 3y − γ| = 6, then α + 2β + 3γ is equal to

Let x = x(t) and y = y(t) be solutions of the differential equations dx/dt + ax = 0 and dy/dt + by = 0 respectively, a, b ∈ R. Given that x(0) = 2, y(0) = 1 and 3y(1) = 2x(1), the value of t, for which x(t) = y(t), is

If the solution curve, of the differential equation dy/dx = (x + y − 2)/(x − y) passing through the point (2, 1) is tan⁻¹((y−1)/(x−1)) − (1/β) log_e(α + ((y−1)/(x−1))²) = log_e|x−1|, then 5β + α is equal to

If y = y(x) is the solution curve of the differential equation (x² − 4) dy − (y² − 3y) dx = 0, x > 2, y(4) = 3/2 and the slope of the curve is never zero, then the value of y(10) equals

If the solution curve y = y(x) of the differential equation (1 + y²)(1 + log_e x) dx + x dy = 0, x > 0 passes through the point (1, 1) and y(e) = (α − tan(3/2))/(β + tan(3/2)), then α + 2β is

A function y = f(x) satisfies f(x) sin 2x + sin x − (1 + cos² x) f'(x) = 0 with condition f(0) = 0. Then, f(π/2) is equal to

Let f(x) = √lim_r→x { 2r²[(f(r))² − f(x)f(r)]/(r² − x²) − r³ e^f(r)/r } be differentiable in (−∞, 0) ∪ (0, ∞) and f(1) = 1. Then the value of e^a, such that f(a) = 0, is equal to

If sin(y/x) = log_e|x| + α/2 is the solution of the differential equation x cos(y/x) dy/dx = y cos(y/x) + x and y(1) = π/3, then α² is equal to

Let y = y(x) be the solution of the differential equation (1 − x²) dy = [xy + (x³ + 2)√3(1 − x²)] dx, −1 < x < 1, y(0) = 0. If y(1/2) = m/n, m and n are coprime numbers, then m + n is equal to

Let y = y(x) be the solution of the differential equation sec x dy − (2 sin x tan x + y) dx = 0 such that y(0) = 2. Then y(2) is equal to

Let Y = Y(X) be a curve lying in the first quadrant such that the area enclosed by the line Y - y = Y'(x)(X - x) and the co-ordinate axes, where (x, y) is any point on the curve, is always -y² / (2Y'(x)) + 1, Y'(x) ≠ 0. If Y(1) = 1, then 12Y(2) is equal to

The solution curve of the differential equation y dx/dy = x(log_e x − log_e y + 1), x > 0, y > 0 passing through the point (e, 1) is

Let y = y(x) be the solution of the differential equation dy/dx = ((tan x) + y)/(sin x (sec x − sin x tan x)), x ∈ (0, π/2) satisfying the condition y(π/4) = 2. Then, y(π/3) is

The temperature T(t) of a body at time t = 0 is 160°F and it decreases continuously as per the differential equation dT/dt = −k(T − 80), where k is positive constant. If T(15) = 120°F, then T(45) is equal to

Let y = y(x) be the solution of the differential equation sec² x dx + (e^2y tan² x + tan x) dy = 0, 0 < x < π/2, y(π/4) = 0. If y(π/6) = α, then e^8α is equal to ___

If x = x(t) is the solution of the differential equation (t + 1) dx = (2x + (t + 1)⁴) dt, x(0) = 2, then x(1) equals

Let y = y(x) be the solution of the differential equation dy/dx = 2x(x + y³)/(xy − 1), y(0) = 1. Then (1/√2 + y(1/√2))² equals

Let the solution y = y(x) of the differential equation dy/dx − y = 1 + 4 sin x satisfy y(π) = 1. Then y(π/2) + 10 is equal to

If the solution y = y(x) of the differential equation (x⁴ + 2x³ + 3x² + 2x + 2) dy − (2x² + 2x + 3) dx = 0 satisfies y(−1) = −π/4, then y(0) is equal to

Let y = y(x) solve (x + y + 2)² dx = dy, y(0) = −2. Let α, β be the maximum and minimum of y(x) on [0, π/3]. If (3α + π)² + β² = γ + δ√3, γ, δ integers, then γ + δ equals

Let y = y(x) be the solution of the differential equation (x² + 4)² dy + (2x³ y + 8xy − 2) dx = 0. If y(0) = 0, then y(2) is equal to

Let f be a differentiable function in the interval (0, ∞) such that f(1) = 1 and lim_t→x (t² f(x) − x² f(t))/(t − x) = 1 for each x > 0. Then 2f(2) + 3f(3) is equal to

If y = y(x) is the solution of the differential equation dy/dx + 2y = sin 2x, y(0) = 3/4, then y(π/8) is equal to

Let y = y(x) be the solution of the differential equation dy/dx = 2x/(1 + x²)² − y e^{1/(1 + x2)}, y(0) = 0. Then the area enclosed by the curve f(x) = y(x) e^{1/(1 + x2)} and the line y − x = 4 is ____.

The differential equation of the family of circles passing the origin and having center at the line y = x is

A conic C passes through (4, −2). At any point P(x, y), x ≥ 3, on C, the tangent slope is half the slope of the line joining P to (3, −5). If the focal distance of (7, 1) on C is d, then 12d equals ____.

Let y = y(x) be the solution of the differential equation (2 + x log_e x) dy/dx + 2y = 3x log_e x, x > 0 and y(e⁻¹) = 0. Then, y(e) is equal to ____.

Let y = y(x) be the solution of the differential equation (1 + x²) dy/dx + y = e^{tan−1 x}, y(1) = 0. Then y(0) is ____.

Let y = y(x) be the solution of the differential equation (1 + y²) e^{tan x} dx − cos² x (1 + e^{2 tan x}) dy = 0, y(0) = 1. Then y(π/4) is equal to

Let f(x) be positive such that the area bounded by y = f(x), y = 0, from x = 0 to x = a > 0 is e^−a + 4a² + a − 1. The differential equation whose general solution is y = c₁f(x) + c₂ is

The solution of the differential equation (x² + y²) dx − 5xy dy = 0, y(1) = 0 is

The solution curve of the differential equation 2y dy/dx + 3 = 5 dy/dx passing through the point (0, 1) is a conic whose vertex lies on the line

For a differentiable function f: R → R, suppose f'(x) = 3 f(x) + α, where α ∈ R, f(0) = 1 and lim_x→−∞ f(x) = 7. Then 9 f(−log_e 3) is equal to

Let ∫₀^x √1 − (y'(t))² dt = ∫₀^x y(t) dt, 0 ≤ x ≤ 3, y ≥ 0, y(0) = 0. Then at x = 2, y'' + y + 1 is equal to :

If x = f(y) is the solution of the differential equation (1 + y²) + (x - 2e^{(tan-1 y)}) dy/dx = 0, y ∈ (-π/2, π/2) with f(0) = 1, then f(1/√3) is equal to:

Let a curve y = f(x) pass through the points (0, 5) and (log_e 2, k). If the curve satisfies the differential equation 2(3 + y) e^(2x) dx - (7 + e^(2x)) dy = 0, then k is equal to:

Let x = x(y) be the solution of the differential equation y = (x - y dx/dy) sin(x/y), y > 0 and x(1) = π/2. Then cos(x(2)) is equal to:

Let y = y(x) be the solution of the differential equation (xy - 5x² √1 + x²) dx + (1 + x²) dy = 0, y(0) = 0. Then y(√3) is equal to:

Let y = y(x) be the solution of the differential equation 2 cos x dy/dx + sin 2x - 4y sin x = 0, x ∈ (0, π/2). If y(π/3) = 0, then y'(π/4) + y(π/4) is equal to:

If y = y(x) is the solution of the differential equation √4 - x² dy/dx = ((sin^-1(x/2))² - y) sin^-1(x/2), -2 ≤ x ≤ 2, y(2) = (π² - 8)/4, then y²(0) is equal to:

Let y = y(x) be the solution of the differential equation cos x (log_e (cos x))² dy + (sin x - 3y sin x log_e (cos x)) dx = 0, x ∈ (0, π/2). If y(π/4) = -1/log_e 2, then y(π/6) is equal to:

If for the solution curve y = f(x) of the differential equation dy/dx + (tan x) y = (2 + sec x)/(1 + 2 sec x)², x ∈ (-π/2, π/2), f(π/3) = √3/10, then f(π/4) is equal to:

Let f: R → R be a thrice differentiable odd function satisfying f'(x) ≥ 0, f''(x) = f(x), f(0) = 0, f'(0) = 3. Then 9f(log_e 3) is equal to ______.

Let y = y(x) be the solution of the differential equation dy/dx + 2y sec² x = 2 sec² x + 3 tan x · sec² x such that y(0) = 5/4. Then 12(y(π/4) - e^(-2)) is equal to ____.

Let g be a differentiable function such that ∫₀^x g(t) dt = x - ∫₀^x t g(t) dt, x ≥ 0. Let y = y(x) satisfy the differential equation dy/dx - y tan x = 2(x + 1) sec x g(x), x ∈ [0, π/2). If y(0) = 0, then y(π/3) is equal to:

Let y = y(x) be the solution of the differential equation dy/dx + 3(tan² x) y + 3y = sec² x, y(0) = 1/3 + e³. Then y(π/4) is equal to:

Let f: [0, ∞) → R be a differentiable function such that f(x) = 1 - 2x + ∫₀^x e^(x-t) f(t) dt for all x ∈ [0, ∞). Then, the area of the region bounded by y = f(x) and the coordinate axes is:

If a curve y = y(x) passes through the point (1, π/2) and satisfies the differential equation (7x⁴ cot y - e^x cosec y) dx/dy = x⁵, x ≥ 1, then at x = 2, the value of cos y is:

Let y = y(x) be the solution curve of the differential equation x(x² + e^x) dy + (e^x (x - 2) y - x³) dx = 0, x > 0, passing through the point (1, 0). Then y(2) is equal to:

Let y = y(x) be the solution of the differential equation (x² + 1) y² - 2xy = (x⁴ + 2x² + 1) cos x. y(0) = 1. Then ∫_(-2)³ y(x) dx is:

Let f(x) = x - 1 and g(x) = e^x for x ∈ R. If dy/dx = (e^(-2√x) g(f(f(x))) - y/√x), y(0) = 0, then y(1) is:

Let y = y(x) be the solution curve of the differential equation (1 + x²) dy + (y - tan^-1 x) dx = 0, y(0) = 1. Then the value of y(1) is :

Let y = y(x) be the solution of the differential equation sec x dy/dx - 2y = 2 + 3 sin x, x ∈ (-π/2, π/2), y(0) = -7/4. Then y(π/6) is equal to

Let the solution curve of the differential equation x dy - y dx = sqrt(x² + y²) dx, x > 0, y(1) = 0 be y = y(x). Then y(3) is equal to

If y = y(x) satisfies the differential equation 16√x·√9+√x·(4+√9+√x) cos y dy = (1 + 2 sin y) dx, x > 0 and y(256) = π/2. If y(49) = α, then 2 sin α is equal to

Let y = y(x) be the solution of the differential equation x⁴ dy + (4x³y + 2 sin x) dx = 0, x > 0, y(π/2) = 0. Then π⁴ y(π/3) is equal to

If the solution curve y = f(x) of the differential equation (x² - 4) y' = 2x(y² - x(4 - x²)) - 20 passes through the point (3, 1.5), then the local maximum value of f is ____

Let y = y(x) be the solution of the differential equation x dy/dx - sin 2y = x³(2 - x³) cos² y, x ≠ 0. If y(2) = π, then tan(y(1)) is equal to

Let y = y(x) be the solution curve of the differential equation (1 + sin x) dy/dx + (y + 1) cos x = 0, y(0) = 0. If the curve y = y(x) passes through the point (α, -1/2), then a value of α is :

Let x = x(y) be the solution of the differential equation 2y² dx/dy - 2xy + x² = 0, y > 1, x(e) = e. Then x(e²) is equal to

Let y = y(x) be the solution of the differential equation dy/dx = (1 + x + x²)(1 - y + y²), y(0) = 1/2. Then (2y(1) - 1) is equal to

Let f : R → R be such that f(xy) = f(x)f(y), for all x, y ∈ R and f(0) ≠ 0. Let g : [1, ∞) → R be a differentiable function such that x² g(x) = ∫₁ˣ (t² f(t) - t g(t)) dt. Then g(2) is equal to :

Let y = y(x) be the solution of the differential equation x√1 - x² dy + (y√1 - x² - x cos^-1 x) dx = 0, x ∈ (0, 1), lim(x→1^-) y(x) = 1. Then y(1/2) equals :