Advertisements
Advertisements
प्रश्न
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
विकल्प
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 particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 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 x + C : y′ + sin x = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (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)`
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
Solution of the differential equation \[\frac{dy}{dx} + \frac{y}{x}=\sin x\] is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
(x3 − 2y3) dx + 3x2 y dy = 0
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
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:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
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 general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
