Advertisements
Advertisements
प्रश्न
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
उत्तर
We have,
\[\frac{dy}{dx} = - 4x y^2 \]
\[ \Rightarrow \frac{1}{y^2}dy = - 4x dx\]
Integrating both sides, we get
\[\int\frac{1}{y^2}dy = - 4\int x dx\]
\[ \Rightarrow \frac{- 1}{y} = - 2 x^2 + C . . . . . \left( 1 \right)\]
Now,
When `x = 0, y = 1`
\[ \therefore - 1 = 0 + C\]
\[ \Rightarrow C = - 1\]
Putting the value of `C` in (1), we get
\[\frac{- 1}{y} = - 2 x^2 - 1\]
\[ \Rightarrow \frac{1}{y} = 2 x^2 + 1\]
\[ \Rightarrow y = \frac{1}{2 x^2 + 1}\]
APPEARS IN
संबंधित प्रश्न
The solution of the differential equation dy/dx = sec x – y tan x is:
(A) y sec x = tan x + c
(B) y sec x + tan x = c
(C) sec x = y tan x + c
(D) sec x + y tan x = c
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
The number of arbitrary constants in the particular solution of a differential equation of third order are ______.
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
Which of the following differential equations has y = x as one of its particular solution?
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
(x3 − 2y3) dx + 3x2 y dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x-coordinate and the product of the x-coordinate and y-coordinate of that point.
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
Find the differential equation of all non-horizontal lines in a plane.
Solution of differential equation xdy – ydx = 0 represents : ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
