Differential Equations Questions

MathematicsDifferential Equations62 Questions Found

MathematicsHomogeneous Differential EquationsJEE Advanced 1999Moderate

A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. Determine the equation of the curve.

Answer:

x^{2} + y^{2} - 2 x = 0

x = 1

MathematicsVariable Separable MethodJEE Advanced 2001Difficult

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A hemispherical tank of radius 2 metres is initially full of water and has an outlet of $12 cm^{2}$ cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $v (t) = 0.6 2 g h (t)$ , where $v (t)$ and $h (t)$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank.

Answer:

\frac{14 π}{27 g} (10)^{5}

units

MathematicsLinear Differential EquationsJEE Advanced 2004Moderate

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A curve 'C' passes through (2, 0) and the slope at (x, y) as $\frac{( x + 1 ) ^{2} + ( y - 3 )}{x + 1}$ . Find the equation of the curve. Find the area bounded by curve and x-axis in fourth quadrant.

Answer:

y = x^{2} - 2 x

;

4/3

sq. units

MathematicsVariable Separable MethodJEE Advanced 2005Moderate

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If length of tangent at any point on the curve $y = f (x)$ intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

Answer:

ln ∣ \frac{1 - 1 - y ^{2}}{y} ∣ + 1 - y^{2} = \pm x + c

MathematicsLinear Differential EquationsJEE Advanced 2009Moderate

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Let $f : R \to R$ be a continuous function which satisfies $f (x) = \int_{0}^{x} f (t) d t$ . Then the value of $f (ln 5)$ is

Answer:0

MathematicsLinear Differential EquationsJEE Advanced 2010Moderate

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Let $f$ be a real-valued differentiable function on $R$ such that $f (1) = 1$ . If the $y$ -intercept of the tangent at any point $P (x, y)$ on the curve $y = f (x)$ is equal to the cube of the abscissa of $P$ , find the value of $f (- 3)$ .

Answer:9

MathematicsLinear Differential EquationsJEE Advanced 2011Easy

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Let $y^{'} (x) + y (x) g^{'} (x) = g (x) g^{'} (x)$ , $y (0) = 0$ , $x \in R$ , where $f^{'} (x)$ denotes $\frac{df ( x )}{d x}$ and $g (x)$ is a given non-constant differentiable function on $R$ with $g (0) = g (2) = 0$ . Then the value of $y (2)$ is

Answer:0

MathematicsLinear Differential EquationsJEE Advanced 2011Moderate

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Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function such that $f (1) = 2$ . If $6 \int_{1}^{x} f (t) d t = 3 x f (x) - x^{3}$ for all $x \geq 1$ , then the value of $f (2)$ is .........

Answer:6

MathematicsOrder and Degree of a Differential EquationJEE Advanced 1998Easy

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The order of the differential equation whose general solution is given by $y = (C_{1} + C_{2}) cos (x + C_{3}) - C_{4} e^{x + C_{5}}$ , where $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ are arbitrary constants, is

(A)

(B)

(C)

(D)

Answer:C

MathematicsFormation of Differential EquationsJEE Advanced 1999Moderate

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The differential equation representing the family of curves $y^{2} = 2 c (x + c)$ , where $c$ is a positive parameter, is of

* Multiple Correct Options

(A)

order 1

(B)

order 2

(C)

degree 3

(D)

degree 4

Answer:A, C

...7

A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. Determine the equation of the curve.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If length of tangent at any point on the curve y=f(x) intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a continuous function which satisfies f(x)=∫0x​f(t)dt. Then the value of f(ln5) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[1,∞)→[2,∞) be a differentiable function such that f(1)=2. If 6∫1x​f(t)dt=3xf(x)−x3 for all x≥1, then the value of f(2) is .........

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A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. Determine the equation of the curve.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If length of tangent at any point on the curve y=f(x) intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a continuous function which satisfies f(x)=∫0x​f(t)dt. Then the value of f(ln5) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[1,∞)→[2,∞) be a differentiable function such that f(1)=2. If 6∫1x​f(t)dt=3xf(x)−x3 for all x≥1, then the value of f(2) is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The differential equation representing the family of curves y2=2c(x+c​), where c is a positive parameter, is of

If length of tangent at any point on the curve $y = f (x)$ intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

Let $f : R \to R$ be a continuous function which satisfies $f (x) = \int_{0}^{x} f (t) d t$ . Then the value of $f (ln 5)$ is

Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function such that $f (1) = 2$ . If $6 \int_{1}^{x} f (t) d t = 3 x f (x) - x^{3}$ for all $x \geq 1$ , then the value of $f (2)$ is .........

The differential equation representing the family of curves $y^{2} = 2 c (x + c)$ , where $c$ is a positive parameter, is of

If length of tangent at any point on the curve $y = f (x)$ intercepted between the point and the x-axis is of length 1. Find the equation of the curve.

Let $f : R \to R$ be a continuous function which satisfies $f (x) = \int_{0}^{x} f (t) d t$ . Then the value of $f (ln 5)$ is

Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function such that $f (1) = 2$ . If $6 \int_{1}^{x} f (t) d t = 3 x f (x) - x^{3}$ for all $x \geq 1$ , then the value of $f (2)$ is .........

The differential equation representing the family of curves $y^{2} = 2 c (x + c)$ , where $c$ is a positive parameter, is of