Advertisements
Advertisements
प्रश्न
(1 + y + x2 y) dx + (x + x3) dy = 0
Advertisements
उत्तर
\[\left( 1 + y + x^2 y \right)dx + \left( x + x^3 \right)dy = 0\]
\[ \Rightarrow dx + y\left( 1 + x^2 \right)dx + x\left( 1 + x^2 \right)dy = 0\]
\[ \Rightarrow dx + \left( 1 + x^2 \right) \left[ ydx + xdy \right] = 0\]
\[ \Rightarrow \left( 1 + x^2 \right) \left[ ydx + xdy \right] = - dx\]
\[ \Rightarrow \left[ ydx + xdy \right] = - \frac{1}{\left( 1 + x^2 \right)}dx\]
\[ \Rightarrow \left[ ydx + xdy \right] = - \frac{dx}{\left( 1 + x^2 \right)}\]
On integrating both side we get,
\[\left( xy \right) = - \int\frac{1}{1 + x^2}dx\]
\[ \Rightarrow xy = - \tan^{- 1} x + c\]
\[ \Rightarrow xy + \tan^{- 1} x = c\]
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=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 – cos y = x : (y sin y + cos y + x) y′ = y
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
If y = etan x+ (log x)tan x then find dy/dx
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The number of arbitrary constants in the general solution of differential equation of fourth order is
Which of the following differential equations has y = x as one of its particular solution?
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
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
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
`(2ax+x^2)(dy)/(dx)=a^2+2ax`
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
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:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[\frac{dy}{dx} + 3y = e^{- 2x}\]
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.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx 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.
If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.
