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Define a Differential Equation. - Mathematics

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प्रश्न

Define a differential equation.

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उत्तर

Differential equation:
An equation containing an independent variable, a dependent variable and differential coefficients of the dependent variable with respect to the independent variable is called a differential equation.
for example: \[\frac{dy}{dx} = e^{x + y}\]

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अध्याय 22: Differential Equations - Very Short Answers [पृष्ठ १३७]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 22 Differential Equations
Very Short Answers | Q 1 | पृष्ठ १३७

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संबंधित प्रश्न

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

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

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


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

 


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


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

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

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

\[\frac{dy}{dx} = \sin^2 y\]

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

Solve the following differential equation: 
(xy2 + 2x) dx + (x2 y + 2y) dy = 0


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

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

Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\]  given that y = 1, when x = 0.


x2 dy + y (x + y) dx = 0


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

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

Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]


Solve the following initial value problem:-

\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]


In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]


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 rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.


The slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).


The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting


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 ______.


Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].


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.
y = 1 − logx `x^2(d^2y)/dx^2 = 1`

Solve the following differential equation.

`x^2 dy/dx = x^2 +xy - y^2`


The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.


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


Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`


Choose the correct alternative:

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


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`


Solve the differential equation `"dy"/"dx" + 2xy` = y


The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:


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