Advertisements
Advertisements
प्रश्न
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Advertisements
उत्तर
Given differential equation is (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0
⇒ `2y(1 + x^2)"d"y = -(1 + y^2) . tan^-1x . "d"x`
⇒ `(2y)/(1 + y^2) "d"y = (tan^-1x)/(1 + x^2) . "d"x`
Integrating both sides, we get
`int (2y)/(1 + y^2) "d"y = -int (tan^-1x)/(1 + x^2) . "d"x`
⇒ `log|1 + y^2| = - 1/2(tan^-1x)^2 + "c"`
⇒ `1/2 (tan^-1x)^2 + log|1 + y^2|` = c
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=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(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 = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
\[\frac{dy}{dx} - y \tan x = e^x\]
(x3 − 2y3) dx + 3x2 y dy = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
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:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the following differential equation:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
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.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
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.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
y = aemx+ be–mx satisfies which of the following differential equation?
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
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.
