Advertisements
Advertisements
प्रश्न
A population grows at the rate of 5% per year. How long does it take for the population to double?
Advertisements
उत्तर
Let P0 be the initial population and P be the population at any time t. Then,
\[\frac{dP}{dt} = \frac{5P}{100}\]
\[ \Rightarrow \frac{dP}{dt} = 0 . 05P\]
\[\Rightarrow \frac{dP}{P} = 0 . 05dt \]
Integrating both sides with respect to t, we get
\[\int\frac{dP}{P} = \int0 . 05dt \]
\[\log P = 0 . 05t + C\]
Now,
\[P = P_0\text{ at }t = 0 \]
\[ \therefore \log P_0 = 0 + C\]
\[ \Rightarrow C = \log P_0 \]
Putting the value of C, we get
\[\log P = 0 . 05t + \log P_0 \]
\[ \Rightarrow \log\frac{P}{P_0} = 0 . 05t\]
To find the time when the population will double, we have
\[P = 2 P_0 \]
\[ \therefore \log\frac{2 P_0}{P_0} = 0 . 05t\]
\[ \Rightarrow \log 2 = 0 . 05t\]
\[ \Rightarrow t = \frac{\log 2}{0 . 05} = 20 \log 2\text{ years }\]
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
The differential equation satisfied by ax2 + by2 = 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 \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
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`
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
`dy/dx = log x`
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Solve the following differential equation y2dx + (xy + x2) dy = 0
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation
`x + y dy/dx` = x2 + y2
