Differential Equations from Integral Equations

JEE Main Mathematics · Differential Equations · 15 questions, latest first

• Let f: R → R be such that f(x y)=f(x) f(y), for all x, y ∈ R and f(0) ≠ 0. Let g:[1, ∞) → R be a differentiable function such…20266 April, Shift 2 · Q18
• Let f:[1, ∞) → R be a differentiable function defined as f(x)=∫_1^x f(t) dt+(1-x)( log_e x-1)+e. Then the value of f(f(1)) is:20265 April, Shift 2 · Q19
• Let f be a twice differentiable function such that f(x)=∫_0^x tan (t-x) d t-∫_0^x f(t) tan t d t, x ∈(-(π)/2, (π)/2). Then f^′…20264 April, Shift 2 · Q25
• Let a differentiable function f satisfy the equation ∫_0^36 f((t x)/36) d t=4 α f(x). If y=f(x) is a standard parabola passing…202624 January, Shift 1 · Q25
• 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(…202623 January, Shift 1 · Q24
• Let f:[1, ∞) → R be a differentiable function. If 6 ∫_1^x f(t) d t=3 x f(x)+x^3-4 for all x ≥ 1, then the value of f(2)-f(3) is202622 January, Shift 1 · Q17
• Let f:[0, ∞) → R be a differentiable function such that f(x)=1-2 x+∫_0^x e^x-t f(t) d t for all x ∈[0, ∞). Then the area of the…20254 April, Shift 1 · Q20
• Let f:[1, ∞) →[2, ∞) be a differentiable function. If 10 ∫_1^x f(t) dt=5 x f(x)-x^5-9 for all x ≥ 1, then the value of f(3) is:20252 April, Shift 2 · Q20
• Let f:(0, ∞) → R be a twice differentiable function. If for some a ≠ 0, ∫_0^1 f(λ x) d λ=a f(x), f(1)=1 and f(16)=1/8, then…202529 January, Shift 1 · Q25
• Let for some function y=f(x), ∫_0^x t f(t) d t=x^2 f(x), x>0 and f(2)=3. Then f(6) is equal to202528 January, Shift 1 · Q18
• 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) d t, x ≥ 0. Then f(2) is equal to ____.202524 January, Shift 1 · Q25
• Let ∫_0^x √(1-(y^′(t))^2) d t=∫_0^x y(t) d t, 0 ≤ x ≤ 3, y ≥ 0, y(0)=0. Then at x=2, y^′ ′+y+1 is equal to20249 April, Shift 2 · Q13
• Let f be a differentiable function such that x^2 f(x)-x=4 ∫_0^x t f(t) d t, f(1)=2/3. Then 18 f(3) is equal to202310 April, Shift 1 · Q9
• Let a differentiable function f satisfy f(x)+∫_3^x (f(t))/t d t=√(x+1), x ≥ 3. Then 12 f(8) is equal to:202331 January, Shift 1 · Q72
• Let f be a differentiable function defined on [0, (π)/2] such that f(x)>0 and f(x)+∫_0^x f(t) √(1-( log_e f(t))^2) d t=e, ∀ x…202324 January, Shift 2 · Q85