Advertisements
Advertisements
प्रश्न
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
पर्याय
y = tan–1x
y – x = k(1 + xy)
x = tan–1y
tan(xy) = k
Advertisements
उत्तर
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is y – x = k(1 + xy).
Explanation:
The given differential equation is `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)`
⇒ `("d"y)/(1 + y^2) = ("d"x)/(1 + x^2)`
Integrating both sides, we get
`int ("d"y)/(1 + y^2) = int ("d"x)/(1 + x^2)`
⇒ tan–1y = tan–1x + c
⇒ tan–1y – tan–1x = c
⇒ `tan^-1((y - x)/(1 + xy))` = c
⇒ `(y - x)/(1 + xy)` = tan c
⇒ `((y - x)/(1 + xy))` = k ....[k = tan c]
⇒ y – x = k(1 + xy)
APPEARS IN
संबंधित प्रश्न
Find the differential equation representing the curve y = cx + c2.
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
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 (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
Find the general solution of `"dy"/"dx" + "a"y` = emx
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 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.
