Linear Differential Equations

JEE Main Mathematics · Differential Equations · 52 questions, latest first

• Let y=y(x) be the solution of the differential equation x √(1-x^2) d y+(y √(1-x^2)-x cos^-1 x) d x=0, x ∈(0,1), lim_x → 1^-…20268 April, Shift 2 · Q17
• Let y=y(x) be the solution of the differential equation (x^2-x √(x^2-1)) d y+(y(x-√(x^2-1))-x) d x=0, x ≥ 1. If y(1)=1, then the…20266 April, Shift 1 · Q25
• Let y=y(x) be the solution of the differential equation ( tan x)^1 / 2 d y=( sec^3 x-( tan x)^3 / 2 y) d x, 0<x<(π)/2,…20265 April, Shift 2 · Q25
• Let y=y(x) be the solution of the differential equation: (d y)/(d x)+((6 x^2+(3 x^2+2 x^3+4) e^-2 x)/((x^3+2)(2+e^-2 x)))…20264 April, Shift 2 · Q19
• Let y=y(x) be the solution of the differential equation x (d y)/(d x)-y=x^2 cot x, x ∈(0, π). If y((π)/2)=(π)/2, then 6…202628 January, Shift 2 · Q19
• Let y=y(x) be the solution of the differential equation x (d y)/(d x)- sin 2 y=x^3(2-x^3) cos^2 y, x ≠ 0. If y(2)=0, then tan…202628 January, Shift 1 · Q19
• Let y=y(x) be a differentiable function in the interval (0, ∞) such that y(1)=2, and lim_t → x((t^2 y(x)-x^2 y(t))/(x-t))=3 for…202624 January, Shift 2 · Q20
• If the solution curve y=f(x) of the differential equation (x^2-4) y^′-2 x y+2 x(4-x^2)^2=0, x>2, passes through the point (3,15),…202623 January, Shift 2 · Q25
• Let y=y(x) be the solution of the differential equation sec x (d y)/(d x)-2 y=2+3 sin x, x ∈(-(π)/2, (π)/2), y(0)=-7/4. Then…202621 January, Shift 2 · Q19
• Let y=y(x) be the solution curve of the differential equation (1+x^2) d y+(y- tan^-1 x) d x=0, y(0)=1. Then the value of y(1) is:202621 January, Shift 1 · Q19
• Let f(x)=x-1 and g(x)=e^x for x ∈ R. If (d y)/(d x)=(e^-2 √x g(f(f(x)))-y/(√x)), y(0)=0, then y(1) is20258 April, Shift 2 · Q20
• Let y=y(x) be the solution of the differential equation (x^2+1) y^′-2 x y=(x^4+2 x^2+1) cos x, y(0)=1. Then ∫_-3^3 y(x) d x is:20257 April, Shift 2 · Q19
• Let y=y(x) be the solution curve of the differential equation x(x^2+e^x) d y+(e^x(x-2) y-x^3) d x=0, x>0, passing through the…20257 April, Shift 1 · Q20
• If a curve y=y(x) passes through the point (1, (π)/2) and satisfies the differential equation (7 x^4 cot y-e^x cosec y) (d x)/(d…20254 April, Shift 2 · Q20
• Let y=y(x) be the solution of the differential equation (d y)/(d x)+3( tan^2 x) y+3 y= sec^2 x, y(0)=1/3+e^3. Then y((π)/4) is…20253 April, Shift 2 · Q20
• Let g be a differentiable function such that ∫_0^x g(t) d t=x-∫_0^x t g(t) d t, x ≥ 0 and let y=y(x) satisfy the differential…20253 April, Shift 1 · Q20
• Let y=y(x) be the solution of the differential equation (d y)/(d x)+2 y sec^2 x=2 sec^2 x+3 tan x · sec^2 x such that y(0)=5/4.…20252 April, Shift 2 · Q25
• If for the solution curve y=f(x) of the differential equation (d y)/(d x)+( tan x) y=(2+ sec x)/((1+2 sec x)^2), x ∈((-π)/2,…202529 January, Shift 2 · Q20
• Let y=y(x) be the solution of the differential equation cos x( log_e( cos x))^2 d y+( sin x-3 y sin x log_e( cos x)) d x=0, x…202529 January, Shift 1 · Q20
• If y=y(x) is the solution of the differential equation, √(4-x^2) (d y)/(d x)=(( sin^-1(x/2))^2-y) sin^-1(x/2),-2 ≤ x ≤ 2,…202528 January, Shift 2 · Q25
• Let f:(0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x^2 f^′(x)=2 x…202524 January, Shift 2 · Q20
• Let y=y(x) be the solution of the differential equation 2 cos x (d y)/(d x)= sin 2 x-4 y sin x, x ∈(0, (π)/2). If y((π)/3)=0,…202524 January, Shift 2 · Q24
• Let y=y(x) be the solution of the differential equation (x y-5 x^2 √(1+x^2)) d x+(1+x^2) d y=0, y(0)=0. Then y(√3) is equal to202524 January, Shift 1 · Q20
• If x=f(y) is the solution of the differential equation (1+y^2)+(x-2 e^tan^-1 y) (d y)/(d x)=0, y ∈(-(π)/2, (π)/2) with f(0)=1,…202522 January, Shift 2 · Q19
• Let y=f(x) be the solution of the differential equation (d y)/(d x)+(x y)/(x^2-1)=(x^6+4 x)/(√(1-x^2)),-1<x<1 such that f(0)=0.…202522 January, Shift 2 · Q23
• Let x=x(y) be the solution of the differential equation y^2 d x+(x-1/y) d y=0. If x(1)=1, then x(1/2) is:202522 January, Shift 1 · Q20
• Let y=y(x) be the solution curve of the differential equation sec y (d y)/(d x)+2 x sin y=x^3 cos y, y(1)=0. Then y(√3) is equal…20248 April, Shift 2 · Q13
• Let y=y(x) be the solution of the differential equation (1+x^2) (d y)/(d x)+y=e^tan^-1 x, y(1)=0. Then y(0) is20246 April, Shift 1 · Q13
• Let y=y(x) be the solution of the differential equation (2 x log_e x) (d y)/(d x)+2 y=3/x log_e x, x>0 and y(e^-1)=0. Then, y(e)…20246 April, Shift 1 · Q14
• Let y=y(x) be the solution of the differential equation (d y)/(d x)+(2 x)/((1+x^2)^2) y=x e^(1/((1+x^2))); y(0)=0. Then the area…20245 April, Shift 2 · Q26
• If y=y(x) is the solution of the differential equation (d y)/(d x)+2 y= sin (2 x), y(0)=3/4, then y((π)/8) is equal to:20245 April, Shift 1 · Q12
• Let f be a differentiable function in the interval (0, ∞) such that f(1)=1 and lim_t → x (t^2 f(x)-x^2 f(t))/(t-x)=1 for each…20245 April, Shift 1 · Q26
• Let y=y(x) be the solution of the differential equation (x^2+4)^2 d y+(2 x^3 y+8 x y-2) d x=0. If y(0)=0, then y(2) is equal to20244 April, Shift 2 · Q12
• Let the solution y=y(x) of the differential equation (d y)/(d x)-y=1+4 sin x satisfy y(π)=1. Then y((π)/2)+10 is equal to ____.20244 April, Shift 1 · Q26
• If (d x)/(d y)=(1+x-y^2)/y, x(1)=1, then 5 x(2) is equal to ____.20241 February, Shift 2 · Q27
• If x=x(t) is the solution of the differential equation (t+1) d x=(2 x+(t+1)^4) dt, x(0)=2, then, x(1) equals ____.20241 February, Shift 1 · Q28
• Let y=y(x) be the solution of the differential equation (d y)/(d x)=(( tan x)+y)/(sin x( sec x- sin x tan x)), x ∈(0, (π)/2)…202431 January, Shift 1 · Q11
• Let y=y(x) be the solution of the differential equation (1-x^2) d y=[x y+(x^3+2) √(3(1-x^2))] d x, -1<x<1, y(0)=0. If y(1/2)=m/n,…202430 January, Shift 1 · Q26
• A function y=f(x) satisfies f(x) sin 2 x+ sin x-(1+ cos^2 x) f^′(x)=0 with condition f(0)=0. Then, f((π)/2) is equal to202429 January, Shift 1 · Q13
• Let x=x(y) be the solution of the differential equation 2(y+2) log_e(y+2) d x+(x+4-2 log_e(y+2)) d y=0, y>-1 with x(e^4-2)=1.…202315 April, Shift 1 · Q11
• If y=y(x) is the solution of the differential equation (d y)/(d x)+(4 x)/((x^2-1)) y=(x+2)/((x^2-1)^(5/2)), x>1 such that…202313 April, Shift 2 · Q27
• Let y=y(x) be the solution of the differential equation (d y)/(d x)+5/(x(x^5+1)) y=((x^5+1)^2)/x^7, x>0. If y(1)=2, then y(2) is…202311 April, Shift 2 · Q13
• Let the solution curve x=x(y), 0<y<(π)/2, of the differential equation ( log_e( cos y))^2 cos y d x-(1+3 x log_e( cos y)) sin y d…20238 April, Shift 2 · Q28
• If the solution curve of the differential equation (y-2 log_e x) d x+(x log_e x^2) d y=0, x>1 passes through the points (e, 4/3)…20238 April, Shift 1 · Q25
• If the solution curve f(x, y)=0 of the differential equation (1+ log_e x) (d x)/(d y)-x log_e x=e^y, x>0, passes through the…20236 April, Shift 2 · Q12
• Let y=y(x) be a solution of the differential equation (x cos x) d y+(x y sin x+y cos x-1) d x=0,0<x<(π)/2. If (π)/3 y((π)/3)=√3,…20236 April, Shift 1 · Q29
• If y=y(x) is the solution curve of the differential equation (d y)/(d x)+y tan x=x sec x, 0 ≤ x ≤ (π)/3, y(0)=1, then y((π)/6) is…20231 February, Shift 1 · Q74
• Let f: R → R be a differentiable function such that f^′(x)+f(x)=∫_0^2 f(t) d t. If f(0)=e^-2, then 2 f(0)-f(2) is equal to ____20231 February, Shift 1 · Q88
• Let the solution curve y=y(x) of the differential equation (d y)/(d x)-(3 x^5 tan^-1(x^3))/((1+x^6)^3 / 2) y=2 x exp {(x^3-…202330 January, Shift 1 · Q70
• Let y=y(x) be the solution of the differential equation x log_e x (d y)/(d x)+y=x^2 log_e x,(x>1). If y(2)=2, then y(e) is equal…202329 January, Shift 2 · Q70
• Let y=y(t) be a solution of the differential equation (d y)/(d t)+α y=γ e^-β t where, α>0, β>0 and γ>0. Then lim_t → ∞ y(t)202325 January, Shift 2 · Q72
• Let y=y(x) be the solution of the differential equation x^3 d y+(x y-1) d x=0, x>0, y(1/2)=3-e. Then y(1) is equal to202324 January, Shift 1 · Q68