Differential Equations

Let $y = f(x)$ be the real valued function defined on the interval $(0, \infty)$, satisfying $y(1) = 0$ and the differential equation\(x \frac{dy}{dx} = y - x^3.\)Then which of the following statements is (are) TRUE?

Let $y : (-\infty, \infty) \to (0, \infty)$ be the solution of the differential equation
\(\frac{dy}{dx} = \frac{e^{5x} y^3 + y^3}{e^x + e^x y^4},\)
satisfying $y(0) = \frac{1}{\sqrt{2}}$. Then the value of $y(\log_e 2)$ is

Let $y(x)$ be the solution of the differential equation
\(x^2 \frac{d y}{d x}+x y=x^2+y^2, \quad x>\frac{1}{e}\)
satisfying $y(1)=0$. Then the value of $2 \frac{(y(e))^2}{y\left(e^2\right)}$ is ____________.

For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying
$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $ $ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $ $ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3}...

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and

\(\lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1\)

for each $x>0$. Then, for all $x>0, f(x)$ is equal to ...

For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation
$\left(x^2-5\right) \frac{d y}{d x}-2 x y=-2 x\left(x^2-5\right)^2$ such that $y(2)=7$.
Then the maximum value of the function $y(x)$ is :

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int\limits_1^x f(t) d t=x f(x)-\frac{x^3}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value o...

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation

\(\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0\)

Then, which of the following statements is/are TRUE ?

If $y(x)$ is the solution of the differential equation

\(x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2,\)

and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is

Let \(\Gamma\) denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on I`, then which of the followin...

Let f : R \(\to\) R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation \({{dy} \over {dx}} = (2 + 5y)(5y - 2)\), then the value of \(\mathop {\lim }\limits_{n \to - \infty } f(x)\) is ...........

If \(g(x) = \int_{\sin x}^{\sin (2x)} {{{\sin }^{ - 1}}} (t)\,dt\), then

If y = y(x) satisfies the differential equation\({8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \right)}^{ - 1}}}\)dx, x > 0 and y(0) = \(\sqrt 7\), then y(256) =

A solution curve of the differential equation
\(\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,\) \(x>0,\) passes through the
point \((1,3)\). Then the solution curve

Let \(f:(0,\infty ) \to R\) be a differentiable function such that \(f'(x) = 2 - {{f(x)} \over x}\) for all \(x \in (0,\infty )\) and \(f(1) \ne 1\). Then

Consider the family of all circles whose centres lie on the straight line \(y=x,\) If this family of circle is represented by the differential equation \(Py'' + Qy' + 1 = 0,\) where \(P, Q\) are functions of \(x,y\) and \(y'\) $$\left( {her...

Let \(y(x)\) be a solution of the differential equation \(\left( {1 + {e^x}} \right)y' + y{e^x} = 1.\)
If \(y(0)=2\), then which of the following statement is (are) true?

The function \(y=f(x)\) is the solution of the differential equation
\({{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,\) in \((-1,1)\) satisfying \(f(0)=0\). Then $$\int\limits_{ - {{\sqrt 3 } \ove...

A curve passes through the point \(\left( {1,{\pi \over 6}} \right)\). Let the slope of
the curve at each point \((x,y)\) be \({y \over x} + \sec \left( {{y \over x}} \right),x > 0.\)
Then the equation of the curve is

If \(y(x)\) satisfies the differential equation \(y' - y\,tan\,x = 2x\,secx\) and \(y(0)=0,\) then

Let \(y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,\) where \(f'(x)\) denotes \({{df\left( x \right)} \over {dx}}\) and \(g(x)\) is a given non-constant differ...

Let \(f:[1,\infty ) \to [2,\infty )\) be a differentiable function such that \(f(1) = 2\). If \(6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5}\) for all \(x \ge 1\), then the value of f(2) is ___________.

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

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...

Match the statements/expressions in Column I with the open intervals in Column II :

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Let a solution \(y=y(x)\) of the differential equation,
\(x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0\) satify \(y\left( 2 \right) = {2 \over {\sqrt 3 }}.\)
STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - ...

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}\) determines a family of circles with :

The solution of primitive integral equation \(\left( {{x^2} + {y^2}} \right)dy = xy\)
\(dx\) is \(y=y(x),\) If \(y(1)=1\) and \(\left( {{x_0}} \right) = e\), then \({{x_0}}\) is equal to

The differential equation \({{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}\) determines a family of circles with

For the primitive integral equation \(ydx + {y^2}dy = x\,dy;\)
\(x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,\) then \(y(-3)\) is

If \(y=y(x)\) and it follows the relation \(x\cos \,y + y\,cos\,x = \pi\) then \(y''(0)=\)

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

If \(y=y(x)\) and \({{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,\)
then \(y\left( {{\pi \over 2}} \right)\) equals

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

If \(y(t)\) is a solution of \(\left( {1 + t} \right){{dy} \over {dt}} - ty = 1\) and \(y\left( 0 \right) = - 1,\) then \(y(1)\) is equal to

A right circular cone with radius \(R\) and height \(H\) contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant \(= k > 0\). Find the time after which the come is empty.

A hemispherical tank of radius \(2\) metres is initially full of water and has an outlet of \(12\) cm2 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$...

If \({x^2} + {y^2} = 1,\) then

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

A solution of the differential equation
\({\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0\) is

The order of the differential equation whose general solution is given by
\(y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},\) where
\({C_1},{C_2},{C_3},{C_4},{C_5},\) are arbitrary constants, is

Let \(u(x)\) and \(v(x)\) satisfy the differential equation \({{du} \over {dx}} + p\left( x \right)u = f\left( x \right)\) and \({{dv} \over {dx}} + p\left( x \right)v = g\left( x \right),\) where \(p(x) f(x)\) and \(g(x)\) are continuous f...

Determine the equation of the curve passing through the origin, in the form \(y=f(x),\) which satisfies the differential equation \({{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,\)

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 \(2.\) From the differential equation and determine a...

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 = {{dy} \over {dx}} = \pm \sqrt {{k^2} - {y...

If \(\left( {a + bx} \right){e^{y/x}} = x,\) then prove that \({x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}\)