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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

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

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प्रश्न

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

पर्याय

  • 4 hours

  • 6 hours

  • 8 hours

  • 10 hours

MCQ
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उत्तर

6 hours

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 2.6: Differential Equations - MCQ

व्हिडिओ ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्‍न

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

Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].

 


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


Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]


For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x\frac{dy}{dx} = y\]
y = ax

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[x^3 \frac{d^2 y}{d x^2} = 1\]
\[y = ax + b + \frac{1}{2x}\]

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

Function y = log x


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


\[\frac{1}{x}\frac{dy}{dx} = \tan^{- 1} x, x \neq 0\]

\[\frac{dy}{dx} = x^5 \tan^{- 1} \left( x^3 \right)\]

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

\[\frac{dy}{dx} = x e^x - \frac{5}{2} + \cos^2 x\]

(1 + x2) dy = xy dx


x cos y dy = (xex log x + ex) dx


\[\sqrt{1 + x^2 + y^2 + x^2 y^2} + xy\frac{dy}{dx} = 0\]

tan y dx + sec2 y tan x dy = 0


Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]


Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]

 


Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.


\[\frac{dy}{dx} + 1 = e^{x + y}\]

y ex/y dx = (xex/y + y) dy


(y2 − 2xy) dx = (x2 − 2xy) dy


Solve the following initial value problem:-

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


Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.


Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is


Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]


Which of the following differential equations has y = C1 ex + C2 ex as the general solution?


The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.


If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]


In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(a^2-x^2)`              `x+y(dy/dx)=0`


Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0


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

Solution D.E.
y = ex  `dy/ dx= y`

Determine the order and degree of the following differential equations.

Solution D.E.
ax2 + by2 = 5 `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx`

For each of the following differential equations find the particular solution.

`y (1 + logx)dx/dy - x log x = 0`,

when x=e, y = e2.


Solve the following differential equation.

xdx + 2y dx = 0


Solve the following differential equation.

`dy/dx + y = e ^-x`


The solution of `dy/dx + x^2/y^2 = 0` is ______


`xy dy/dx  = x^2 + 2y^2`


y dx – x dy + log x dx = 0


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

The differential equation of y = Ae5x + Be–5x is


Solve the differential equation xdx + 2ydy = 0


Solve: `("d"y)/("d"x) + 2/xy` = x2 


Solve the following differential equation 

sec2 x tan y dx + sec2 y tan x dy = 0

Solution: sec2 x tan y dx + sec2 y tan x dy = 0

∴ `(sec^2x)/tanx  "d"x + square` = 0

Integrating, we get

`square + int (sec^2y)/tany  "d"y` = log c

Each of these integral is of the type

`int ("f'"(x))/("f"(x))  "d"x` = log |f(x)| + log c

∴ the general solution is

`square + log |tan y|` = log c

∴ log |tan x . tan y| = log c

`square`

This is the general solution.


Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.


Solution of `x("d"y)/("d"x) = y + x tan  y/x` is `sin(y/x)` = cx


If `y = log_2 log_2(x)` then `(dy)/(dx)` =


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