English

Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in - Mathematics and Statistics

Advertisements
Advertisements

Question

Select and write the correct alternative from the given option for the question

Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in

Options

  • 4 hours

  • 6 hours

  • 8 hours

  • 10 hours

MCQ

Solution

6 hours

shaalaa.com
  Is there an error in this question or solution?
Chapter 2.6: Differential Equations - MCQ

Video TutorialsVIEW ALL [2]

RELATED QUESTIONS

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`


\[\frac{d^3 x}{d t^3} + \frac{d^2 x}{d t^2} + \left( \frac{dx}{dt} \right)^2 = e^t\]

\[\frac{d^2 y}{d x^2} + 4y = 0\]

\[x + \left( \frac{dy}{dx} \right) = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.


Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]


Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.


Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]

Function y = ex + 1


Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x


\[\sqrt{a + x} dy + x\ dx = 0\]

\[\frac{dy}{dx} = x \log x\]

\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y\left( 2 \right) = 0\]

\[\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\]

(1 + x2) dy = xy dx


\[\frac{dy}{dx} = \frac{x e^x \log x + e^x}{x \cos y}\]

y (1 + ex) dy = (y + 1) ex dx


\[\frac{dy}{dx} = 1 - x + y - xy\]

dy + (x + 1) (y + 1) dx = 0


\[\left( x - 1 \right)\frac{dy}{dx} = 2 x^3 y\]

\[\frac{dy}{dx} = e^{x + y} + e^{- x + y}\]

Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]

 


\[\frac{dy}{dx} = 2 e^x y^3 , y\left( 0 \right) = \frac{1}{2}\]

Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.


Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\] 

 


\[\frac{dy}{dx}\cos\left( x - y \right) = 1\]

\[\cos^2 \left( x - 2y \right) = 1 - 2\frac{dy}{dx}\]

\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]


3x2 dy = (3xy + y2) dx


\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]

Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.

 


Solve the following initial value problem:-

\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]


Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]


Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]


In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?


The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?


If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.

 

In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]


Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.

 

Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.


A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.


Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\]  are rectangular hyperbola.


Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.


If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.


The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting


y2 dx + (x2 − xy + y2) dy = 0


If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`


Choose the correct option from the given alternatives:

The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of


Choose the correct option from the given alternatives:

The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is


In the following example, verify that the given function is a solution of the corresponding differential equation.

Solution D.E.
y = xn `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0`

Determine the order and degree of the following differential equations.

Solution D.E.
y = 1 − logx `x^2(d^2y)/dx^2 = 1`

Solve the following differential equation.

`dy/dx = x^2 y + y`


Solve the following differential equation.

`y^3 - dy/dx = x dy/dx`


For the following differential equation find the particular solution.

`(x + 1) dy/dx − 1 = 2e^(−y)`,

when y = 0, x = 1


Solve the following differential equation.

xdx + 2y dx = 0


Choose the correct alternative.

The differential equation of y = `k_1 + k_2/x` is


The solution of `dy/ dx` = 1 is ______


Choose the correct alternative.

The integrating factor of `dy/dx -  y = e^x `is ex, then its solution is


Solve the differential equation:

dr = a r dθ − θ dr


Solve:

(x + y) dy = a2 dx


`xy dy/dx  = x^2 + 2y^2`


Solve: `("d"y)/("d"x) + 2/xy` = x2 


Solve the following differential equation

`x^2  ("d"y)/("d"x)` = x2 + xy − y2 


Choose the correct alternative:

Solution of the equation `x("d"y)/("d"x)` = y log y is


A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution


The function y = ex is solution  ______ of differential equation


The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______


State whether the following statement is True or False:

The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x 


The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`


Solve the following differential equation `("d"y)/("d"x)` = x2y + y


Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.


An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.


Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]


The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:


Solve the differential equation

`y (dy)/(dx) + x` = 0


Solve the differential equation

`x + y dy/dx` = x2 + y2


The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×