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

The solution of differential equation x2d2ydx2 = 1 is ______

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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

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


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

\[\cos x\frac{dy}{dx} - \cos 2x = \cos 3x\]

x cos2 y  dx = y cos2 x dy


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

\[\cos x \cos y\frac{dy}{dx} = - \sin x \sin y\]

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

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

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

\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]

3x2 dy = (3xy + y2) dx


Solve the following initial value problem:-

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


A population grows at the rate of 5% per year. How long does it take for the population to double?


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?


Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\]  at any point (x, y) on it.


At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.


The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when


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


Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.


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`


A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.


y2 dx + (xy + x2)dy = 0


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

Differential equation of the function c + 4yx = 0 is


Solve: `("d"y)/("d"x) + 2/xy` = x2 


For the differential equation, find the particular solution

`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0


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


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

Solution: `("d"y)/("d"x)` = cos(x + y)    ......(1)

Put `square`

∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`

∴ (1) becomes `"dv"/("d"x) - 1` = cos v

∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

Integrating, we get

`int 1/(1 + cos "v")  "d"v = int  "d"x`

∴ `int 1/(2cos^2 ("v"/2))  "dv" = int  "d"x`

∴ `1/2 int square  "dv" = int  "d"x`

∴ `1/2* (tan("v"/2))/(1/2)` = x + c

∴ `square` = x + c


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