Differential Equations Questions

MathematicsDifferential Equations62 Questions Found

MathematicsVariable Separable MethodJEE Advanced 2006Difficult

Match the following :

List-I

(P)

\int_{0}^{π /2} (sin x)^{c o s x} (cos x cot x - lo g (sin x)^{s i n x}) d x

(Q)

Area bounded by

- 4 y^{2} = x

and

x - 1 = - 5 y^{2}

(R)

Cosine of the angle of intersection of curves

y = 3^{x - 1} lo g x

and

y = x^{x} - 1

(S)

Let

\frac{d y}{d x} = \frac{6}{x + y}

where

y (0) = 0

then value of

y

when

x + y = 6

List-II

(1)

(2)

(3)

6 ln 2

(4)

4/3

Answer:P → 1, Q → 4, R → 1, S → 3

MathematicsLinear Differential EquationsJEE Advanced 2009Difficult

View in:English Hindi

Match the statements/expressions given in Column-I with the values given in Column-II.

List-I

(P)

The number of solutions of the equation

x e^{s i n x} - cos x = 0

in the interval

(0, \frac{π}{2})

(Q)

Value(s) of

k

for which the planes

k x + 4 y + z = 0

4 x + k y + 2 z = 0

and

2 x + 2 y + z = 0

intersect in a straight line

(R)

Value(s) of

k

for which

∣ x - 1∣ + ∣ x - 2∣ + ∣ x + 1∣ + ∣ x + 2∣ = 4 k

has integer solution(s)

(S)

y^{'} = y + 1

and

y (0) = 1

, then value(s) of

y (ln 2)

List-II

(1)

(2)

(3)

(4)

(5)

Answer:P → 1, Q → 4, R → NaN, S → 3

MathematicsLinear Differential EquationsJEE Advanced 1997Moderate

View in:English Hindi

Let $u (x)$ and $v (x)$ satisfy the differential equation $\frac{d u}{d x} + p (x) u = f (x)$ and $\frac{d v}{d x} + p (x) v = g (x)$ , where $p (x), f (x)$ and $g (x)$ are continuous functions. If $u (x_{1}) > v (x_{1})$ for some $x_{1}$ and $f (x) > g (x)$ for all $x > x_{1}$ , prove that any point $(x, y)$ where $x > x_{1}$ , does not satisfy the equations $y = u (x)$ and $y = v (x)$ .

MathematicsLinear Differential EquationsJEE Advanced 2000Moderate

View in:English Hindi

A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to $\frac{l n 10 - l n 9}{l n ( 1.04 ) - 0.03}$ .

MathematicsLinear Differential EquationsJEE Advanced 2001Moderate

View in:English Hindi

Let $f (x), x \geq 0$ , be a non-negative continuous function, and let $F (x) = \int_{0}^{x} f (t) d t, x \geq 0$ . If for some $c > 0, f (x) \leq c F (x)$ for all $x \geq 0$ , then show that $f (x) = 0$ for all $x \geq 0$ .

MathematicsVariable Separable MethodJEE Advanced 1994Moderate

View in:English Hindi

A normal is drawn at a point $P (x, y)$ of a curve. It meets the x-axis at Q. If PQ is of constant length $k$ , then show that the differential equation describing such curves is $y \frac{d y}{d x} = \pm k^{2} - y^{2}$ . Find the equation of such a curve passing through (0, k).

Answer:

x^{2} + y^{2} = k^{2}

MathematicsFormation of Differential EquationsJEE Advanced 1995Moderate

View in:English Hindi

Let $y = f (x)$ be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.

Answer:

x + y = 2

and

x y = 1

MathematicsVariable Separable MethodJEE Advanced 1996Moderate

View in:English Hindi

Determine the equation of the curve passing through the origin, in the form $y = f (x)$ , which satisfies the differential equation $\frac{d y}{d x} = sin (10 x + 6 y)$ .

Answer:

y = \frac{1}{3} [tan^{- 1} (\frac{5 t a n 4 x}{4 - 3 t a n 4 x}) - 5 x]

MathematicsVariable Separable MethodJEE Advanced 1996Moderate

View in:English Hindi

A curve $y = f (x)$ passes through the point $P (1, 1)$ . The normal to the curve at $P$ is $a (y - 1) + (x - 1) = 0$ . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $y$ -axis, the curve and the normal to the curve at $P$ .

Answer:

y = e^{a (x - 1)}

; 1 sq. unit

MathematicsVariable Separable MethodJEE Advanced 1998Moderate

View in:English Hindi

A curve $C$ has the property that if the tangent drawn at any point $P$ on $C$ meets the co-ordinate axes at $A$ and $B$ , then $P$ is the mid-point of $A B$ . The curve passes through the point $(1, 1)$ . Determine the equation of the curve.

Answer:

x y = 1

...7

Match the following :

List-I

List-II

Match the statements/expressions given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x),x≥0, be a non-negative continuous function, and let F(x)=∫0x​f(t)dt,x≥0. If for some c>0,f(x)≤cF(x) for all x≥0, then show that f(x)=0 for all x≥0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Determine the equation of the curve passing through the origin, in the form y=f(x), which satisfies the differential equation dxdy​=sin(10x+6y).

Match the following :

List-I

List-II

Match the statements/expressions given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x),x≥0, be a non-negative continuous function, and let F(x)=∫0x​f(t)dt,x≥0. If for some c>0,f(x)≤cF(x) for all x≥0, then show that f(x)=0 for all x≥0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Determine the equation of the curve passing through the origin, in the form y=f(x), which satisfies the differential equation dxdy​=sin(10x+6y).

Let $f (x), x \geq 0$ , be a non-negative continuous function, and let $F (x) = \int_{0}^{x} f (t) d t, x \geq 0$ . If for some $c > 0, f (x) \leq c F (x)$ for all $x \geq 0$ , then show that $f (x) = 0$ for all $x \geq 0$ .

Determine the equation of the curve passing through the origin, in the form $y = f (x)$ , which satisfies the differential equation $\frac{d y}{d x} = sin (10 x + 6 y)$ .

Let $f (x), x \geq 0$ , be a non-negative continuous function, and let $F (x) = \int_{0}^{x} f (t) d t, x \geq 0$ . If for some $c > 0, f (x) \leq c F (x)$ for all $x \geq 0$ , then show that $f (x) = 0$ for all $x \geq 0$ .

Determine the equation of the curve passing through the origin, in the form $y = f (x)$ , which satisfies the differential equation $\frac{d y}{d x} = sin (10 x + 6 y)$ .