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महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

The solution of differential equation x2d2ydx2 = 1 is ______ - Mathematics and Statistics

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प्रश्न

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

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उत्तर

y = 1 – log x

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पाठ 1.8: Differential Equation and Applications - Q.2

संबंधित प्रश्‍न

\[\sqrt{1 + \left( \frac{dy}{dx} \right)^2} = \left( c\frac{d^2 y}{d x^2} \right)^{1/3}\]

Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.


Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]

 


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

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

 


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.


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

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

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

(x2 − y2) dx − 2xy dy = 0


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

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?


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?


Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\]  and tangent at any point of which makes an angle tan−1  \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.


Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).


Define a differential equation.


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


Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?


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 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`

For  the following differential equation find the particular solution.

`dy/ dx = (4x + y + 1),

when  y = 1, x = 0


Solve the following differential equation.

`dy/dx + 2xy = x`


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


Solve the following differential equation

`y log y ("d"x)/("d"y) + x` = log y


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.


Find the particular solution of the following differential equation

`("d"y)/("d"x)` = e2y cos x, when x = `pi/6`, y = 0.

Solution: The given D.E. is `("d"y)/("d"x)` = e2y cos x

∴ `1/"e"^(2y)  "d"y` = cos x dx

Integrating, we get

`int square  "d"y` = cos x dx

∴ `("e"^(-2y))/(-2)` = sin x + c1

∴ e–2y = – 2sin x – 2c1

∴ `square` = c, where c = – 2c

This is general solution.

When x = `pi/6`, y = 0, we have

`"e"^0 + 2sin  pi/6` = c

∴ c = `square`

∴ particular solution is `square`


Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.


Solve: ydx – xdy = x2ydx.


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