Linear Differential Equations: JEE Questions & Solutions

Let y = y(x) be the solution curve of the differential equation (1+x^2)dy+(y- ^-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)m^2-2f'(x) m+f''(x) = 0 in m, has two equal roots for…

Let y = y(x) be the solution of the differential equation x dydx - 2y = 2 + 3 x, x (- 2, 2). If y(0) = - 74, then y ( 6) is equal to:

Let f: [1,) R be a differentiable function. If 6 1^x f(t) dt = 3x f(x) + x^3 - 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 x^2 dy + (4x^2 y + 2 x)dx = 0, x 0, y ( 2) = 0. Then ^4 y ( 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( 2(1)), f(…

If the solution curve y = f(x) of the differential equation (x^2 - 4) y' - 2xy + 2x(4 - x^2)^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 t x ( t^2y(x)-x^2y(t)x-t)=3 for each x 0. Then 2y(2)…

Let y = y(x) be the solution of the differential equation x dydx - 2y = x^3(2 - x^3) ^2 y, x 0. If y(2) = 0, then (y(1)) is equal to:

Let y=y(x) be the solution of the differential equation x dydx=y-x^2 x, x (0,) If y ( 2)= ^22, then 6y ( 6)-8y ( 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 y^2 dx + ( x - 1y) dy = 0. If x(1) = 1, then x ( 12) 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 n=1^100 e f(n) is…

If x = f(y) is the solution of the differential equation (1 + y^2) + (x - 2e^ ^-1y) dydx = 0, y ( - 2, 2), with f(0) = 1, then f ( 1 3) is…

Let y = f(x) be the solution of the differential equation dydx + xyx^2 - 1 = x^6 + 4x 1 - x^2, -1 < x < 1 such that f(0) = 0. If 6 -1/2^1/2…

Let y = y(x) be the solution of the differential equation ( xy - 5x^2 1 + x^2) dx + (1 + x^2) dy = 0, y(0) = 0. Then y( 3) is equal to:

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

Let for some function y = f(x), 0^x t f(t) dt = x^2 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 - x^2 dydx = ( ( ^-1 ( x2))^2 - y) ^-1 ( x2) where -2 x 2, and y(2) = ^2 - 84,…

Let y = y(x) be the solution of the differential equation x ( e ( x))^2 dy + ( x - 3 y x e ( x)) dx = 0, x ( 0, 2) If y ( 4) = -1, then y (…

Let f: (0,) R be a twice differentiable function. If for some a 0, 0^a f(x) dx = f(a), f(1) = 1, f(16) = 18, then 16 - f^-1 ( 116) is equal…

If for the solution curve y = f(x) of the differential equation dydx + ( x) y = 2 + ^2 x, y ( 3) = 3, then y ( 4) is equal to:

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

Let y = y(x) be the solution of the differential equation dydx + 2y ^2 x = 2 ^2 x + 3 x ^2 x such that y(0) = 54. Then 12 ( y ( 4) - e^2)…

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 dydx -…

Let y = y(x) be the solution of the differential equation dydx + 3( ^2 x) y + 3y = ^2 x, with y(0) = 13 + e^3. Then y ( 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…

If a curve y = y(x) passes through the point (1, 2) and satisfies the differential equation (7x^4 y - e^x y) dxdy = x^5, x 1 then at x = 2,…

Let y = y(x) be the solution curve of the differential equation x(x^2 + e^x) dy + ( e^x(x - 2) y - x^3) dx = 0, x 0, passing through the…

Let y = y(x) be the solution of the differential equation (x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1) x, with the initial condition y(0) = 1. Then…

Let f(x) = x - 1 and g(x) = e^x for x R. If dydx = ( 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 dxdt +ax=0 and dydt +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 x + y + 3 |2x + 3y - | = 6. Find +2…

A function y = f(x) satisfies f(x) (2x) + (x) - (1 + (2x))f'(x) = 0 with f(0) = 0. Then f ( 2) is equal to:

Let f(x)= r x [ 2r^2(f(r^2)-f(x))f(r)r^2-x^2-r^2e^f(r)/r ] be differentiable in (-,0) (0,) and f(1)=1. Then the value of e^a, such that…

Let y = y(x) be the solution of the differential equation (x) dy + 2(1 - x) (x) + x(2 - x) dx = 0 such that y(0) = 2. Then y(2) is equal to:

Let y = y(x) be the solution of the differential equation (1 - x^2)dy = xy + x^3 + 2 1-x^2 dx, with y(0) = 0. If y ( 12) = mn, where m and…

Let f: R - 0 R be a function satisfying f ( xy) = 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 be a non-zero real number. Suppose f: R R is a differentiable function such that f(0) = 2 and x f(x) = 1. If f'(x) = f(x) + 3, then f(-…

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

If (e) y = 3 ^-1 x, then (1 - x)^2 y'' - x y' at x = 12 is equal to:

Suppose the solution of the differential equation dydx = [(2 +)x - y + 2][ x - 2 y - ( y - 4)] represents a circle passing through origin.…

The differential equation, whose general solution is y = c1 f(x) + c2, where c1 and c2 are arbitrary constants, is:

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 dydx = yx(1 + xy^2(1 + x)), x 0, y(1) = 3. Then y^2(x)9 is equal to:

Let y = y(t) be a solution to the differential equation: dydt + y = e^- t where,, 0. We are interested in finding the value of: t y(t) then:

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 e x dydx + y = x^2 e x, (x 1). If y(2) = 2, then y(e) is equal to:

Let the solution curve y = y(x) of the differential equation dydx - 3x^5 ^-1(x^3)(1+x^6)^3/2 y = 2x x^3- ^-1x^3 (1+x)^6 pass through the…

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

If y = y(x) is the solution curve of the differential equation dydx + y x = x x, 0 x 3 and y(0) = 1, then y ( 6) is equal to:

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

Let x = (x^ y^) be the solution of the differential equation 2x^2 y dy - (1 - xy^2) dx = 0, x 0, y(2) = e 2. Then + - equals:

Let y = y(x) be a solution of the differential equation (x x)dy + (xy x + y x - 1)dx = 0, 0 < x < /2. If y( /3) = 3, then |y''( /6) + 2y'(…

If the solution curve of the differential equation (y - 2 x) dx + (x x^2) dy = 0 passes through the points ( e^4/3,) and ( e^4,), then is…

Let f be a differentiable function such that x^2f(x) - x = 4 0^x tf(t) dt, f(1) = 23. Then 18f(3) is equal to:

Let the tangent at any point P on a curve passing through the points (1, 1) and ( 110, 100), intersect positive x-axis and y-axis at the…

Let f: [2, 4] R be a differentiable function such that (x x) f'(x) + ( x) f(x) 1, x [2, 4] with f(2) = 12 and f(4) = 14. Consider the…

Let y = y(x) be the solution of the differential equation dydx + 5x(x^5 + 1)y = (x^5 + 1)^2x^7, 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 + x^2) dy = y(x - y) dx. If y(0) = 1 and y(2 2) =, then:

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

If y = y(x) is the solution of the differential equation dydx = 4x(x - 1)^2 - x^2 - 2(x - 1)^3 such that y(2) = 29 2 ( 2 + 5) and y(x) = (…

Let x=x(y) be the solution of the differential equation 2(y+2) e(y+2) dx+(x+4-2 e(y+2)) dy=0, y-1, with x(e^4-2)=1. Then x(e^9-2) is equal…