Advertisements
Advertisements
प्रश्न
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Advertisements
उत्तर
Given equation can be written as
`x/(1+x^2)dx-y/(1+y^2)dy=0`
Integrating to get
`1/2 log (1+x^2)-1/2log(1+y^2)=logc_1`
`=>log(1+x^2)-log(1+y^2)=logc_1^2=logc`
`therefore (1+x^2)/(1+y^2)=c`
`x=0,y=1=>c=1/2`
`therefore 1+y^2=2(1+x^2) or y=sqrt(2x^2+1)`
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
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.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Solve the differential equation `cos^2 x dy/dx` + y = tan x
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
cos (x + y) dy = dx
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
The solution of differential equation coty dx = xdy is ______.
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 given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
