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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

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

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


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

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

(1 + x2) dy = xy dx


x cos2 y  dx = y cos2 x dy


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

Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.


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

2xy dx + (x2 + 2y2) dy = 0


Solve the following initial value problem:-

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


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


Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]


The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).


Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.


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


The differential equation satisfied by ax2 + by2 = 1 is


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Determine the order and degree of the following differential equations.

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

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


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

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Choose the correct alternative:

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


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General solution of `y - x ("d"y)/("d"x)` = 0 is


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∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

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∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

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`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


The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.


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