Linear Differential Equations: JEE Questions & Solutions

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

Let f: R → R be a twice differentiable function such that the quadratic equation f(x)m²-2f'(x) m+f''(x) = 0 in m, has two equal roots for…

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

Let f: [1,∞) → ℝ be a differentiable function. If 6∫₁^x f(t) dt = 3x f(x) + x³ - 4 for all x ≥ 1, then the value of f(2) - f(3) is

Let y = y(x) be the solution of the differential equation x⁴ dy + (4x³ y + 2sin x)dx = 0, x > 0, y(π/2) = 0. Then π⁴ y(π/3) is equal to:

Let f be a twice differentiable non-negative function such that (f(x))² = 25 + ∫₀^x ( f(t)² + (f'(t))²) dt. Then the mean of f(logₑ(1)),…

If the solution curve y = f(x) of the differential equation (x² - 4) y' - 2xy + 2x(4 - x²)² = 0, x > 2, passes through the point (3, 15),…

Let y = y(x) be a differentiable function in the interval (0,∞) such that y(1) = 2, and lim_(t → x)((t²y(x)-x²y(t))/(x-t))=3 for each x >…

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

Let y=y(x) be the solution of the differential equation xdy/dx-y=x²cot x, x∈(0,π) If y (π/2)=π/2, then 6y (π/6)-8y (π/4) is equal to:

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) - 3a…

Let x = x(y) be the solution of the differential equation y² dx + ( x - 1/y) dy = 0. If x(1) = 1, then x( 1/2) is:

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 Σ_(n=1)¹⁰⁰ logₑ…

If x = f(y) is the solution of the differential equation (1 + y²) + (x - 2e^tan⁻¹y) dy/dx = 0, y ∈ ( -π/2, π/2), with f(0) = 1, then f(…

Let y = f(x) be the solution of the differential equation dy/dx + xy/(x² - 1) = (x⁶ + 4x)/√(1 - x²), -1 < x < 1 such that f(0) = 0. If 6…

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 f be a differentiable function such that 2(x + 2)² f(x) - 3(x + 2)² = 10∫₀^x (t + 2) f(t) dt for x ≥ 0. Then f(2) is equal to.

Let f: (0, ∞) → ℝ be a function which is differentiable at all points of its domain and satisfies the condition x² f'(x) = 2x f(x) + 3,…

Let for some function y = f(x), ∫₀^x t f(t) dt = x² f(x), x > 0 and f(2) = 3. Then f(6) 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) where -2 ≤ x ≤ 2, and y(2)…

Let y = y(x) be the solution of the differential equation cos x (logₑ (cos x))² dy + (sin x - 3 y sin x logₑ (cos x)) dx = 0, x ∈ ( 0,…

Let f: (0, ∞) → ℝ be a twice differentiable function. If for some a ≠ 0, ∫₀¹ f(λ x) dλ = a f(x), f(1) = 1 and f(16) = 1/8, then 16 - f'(…

If for the solution curve y = f(x) of the differential equation dy/dx + (tan x) y = (2 + sec x)/(1 + 2sec x)², x ∈ (-π/2, π/2), f(π/3) =…

Let f: [1, ∞) → [2, ∞) be a differentiable function. If 10∫₁^x f(t) dt = 5x f(x) - x⁵ - 9 for all x ≥ 1, then the value of f(3) is:

Let y = y(x) be the solution of the differential equation dy/dx + 2y sec² x = 2sec² x + 3tan x sec² x such that y(0) = 5/4. Then 12(y(π/4)…

Let g be a differentiable function such that ∫₀^x g(t) dt = x - ∫₀^x t g(t) dt, x ≥ 0 and let y = y(x) satisfy the differential equation…

Let y = y(x) be the solution of the differential equation dy/dx + 3(tan² x) y + 3y = sec² x, with 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…

If a curve y = y(x) passes through the point (1, π/2) and satisfies the differential equation (7x⁴ cot y - e^x csc y) dx/dy = x⁵, x ≥ 1…

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…

Let y = y(x) be the solution of the differential equation (x² + 1)y' - 2xy = (x⁴ + 2x² + 1)cos x, with the initial condition y(0) = 1. Then…

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 x=x(t) and y=y(t) be solutions of dx/dt +ax=0 and dy/dt +by=0 respectively. Given x(0)=2, y(0)=1, and 3y(1)=2x(1), find t for which…

If the solution for Differential equation (2x + 3y - 2)dx + (4x + 6y - 7)dy = 0, y(0) = 3, has solution α x + β y + 3log|2x + 3y - γ| = 6.…

A function y = f(x) satisfies f(x)sin(2x) + sin(x) - (1 + cos² x)f'(x) = 0 with 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)-r³e^(f(r)/r)] be differentiable in (-∞,0)∪(0,∞) and f(1)=1. Then the value of ea, such…

Let y = y(x) be the solution of the differential equation sec(x) dy + {2(1 - x) tan(x) + x(2 - x)} dx = 0 such that y(0) = 2. Then y(2) is…

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) =…

Let f: ℝ - {0} → ℝ be a function satisfying f(x/y) = f(x)/f(y) for all x, y with f(y) ≠ 0. If f'(1) = 2024, then:

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 coordinate axes,…

Let α be a non-zero real number. Suppose f: R → R is a differentiable function such that f(0) = 2 and lim_(x → ∞) f(x) = 1. If f'(x) = α…

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

If logₑ y = 3sin⁻¹ x, then (1 - x²) y'' - x y' at x = 1/2 is equal to:

Suppose the solution of the differential equation dy/dx = [(2 + α)x - β y + 2]/[β x - 2α y - (β y - 4α)] represents a circle passing…

Let f(x) be a positive function such that the area bounded by y = f(x), y = 0, from x = 0 to x = a > 0 is e^(-a) + 4a² + a - 1. Then the…

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:

Let y = y(x) be the solution curve of the differential equation dy/dx = y/x(1 + xy²(1 + log x)), x > 0, y(1) = 3. Then y²(x)/9 is equal to:

Let y = y(t) be a solution to the differential equation: dy/dt + α y = γ e^(-β t) where α, β, γ > 0. We are interested in finding the value…

Let f and g be twice differentiable functions on ℝ such that f''(x) = g''(x) + 6x, f'(1) = 4g'(1) - 3 = 9, and f(2) = 3g(2) = 12. Which of…

Let y = y(x) be the solution of the differential equation x logₑ x dy/dx + y = x² logₑ x, (x > 1). If y(2) = 2, then y(e) is equal to:

Let the solution curve y = y(x) of the differential equation dy/dx - (3x⁵ tan⁻¹(x³))/(1+x⁶)^(3/2) y = 2x exp ((x³ - tan⁻¹(x³))/√(1+x⁶))…

Let a differentiable function f satisfy f(x) + ∫₃^x f(t) dt = √(x+1), x ≥ 3. Then 12 f(8) is equal to:

If y = y(x) is the solution curve of the differential equation dy/dx + y tan x = x sec x, 0 ≤ x ≤ π/3 and y(0) = 1, then y ( π/6) is equal…

Let f: ℝ → ℝ be a differentiable function such that f'(x) + f(x) = ∫₀² f(t) dt. If f(0) = e⁻², then 2f(0) - f(2) is equal to:

Let α x = exp(x^β y^γ) be the solution of the differential equation 2x² y dy - (1 - xy²) dx = 0, x > 0, y(2) = √(lnₑ 2). Then α + β - γ…

Let y = y(x) be a solution of the differential equation (xcos x) dy + (xysin x + ycos x - 1) dx = 0, 0 < x < π/2. If π/3 y (π/3) = √3, then…

If the solution curve of the differential equation (y - 2logₑ x)dx + (x logₑ x²)dy = 0, x > 1, passes through the points (e, 4/3) and (e⁴,…

Let f be a differentiable function such that x²f(x) - x = 4∫₀^x tf(t) dt, f(1) = 2/3. Then 18f(3) is equal to:

Let the tangent at any point P on a curve passing through the points (1, 1) and ( 1/10, 100), intersect positive x-axis and y-axis at the…

Let f: [2, 4] → ℝ be a differentiable function such that (x logₑ x) f'(x) + (logₑ x) f(x) + f(x) ≥ 1 for all x ∈ [2, 4], with f(2) = 1/2…

Let y = y(x) be the solution of the differential equation dy/dx + 5/x(x⁵ + 1)y = (x⁵ + 1)²/x⁷, x > 0. If y(1) = 2, then y(2) is equal to:

Let y = y(x), y > 0, be a solution curve of the differential equation (1 + x²) dy = y(x - y) dx. If y(0) = 1 and y(2√2) = β, then:

Let y = y₁(x) and y = y₂(x) be the solution curves of the differential equation dy/dx = y + 7 with initial conditions y₁(0) = 0 and y₂(0) =…

If y = y(x) is the solution of the differential equation dy/dx + 4x/(x² - 1) y = (x + 2)/(x² - 1)^(5/2), x > 1, such that y(2) = 2/9logₑ(2…

Let x=x(y) be the solution of the differential equation 2(y+2)logₑ(y+2) dx+(x+4-2logₑ(y+2)) dy=0, y>-1, with x(e⁴-2)=1. Then x(e⁹-2) is…