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
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.
Solve the differential equation `cos^2 x dy/dx` + y = tan x
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The solution of the differential equation \[x\frac{dy}{dx} = y + x \tan\frac{y}{x}\], is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
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 \[\frac{dy}{dx} = e^{x + y}\], is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.
\[\frac{dy}{dx} = \left( x + y \right)^2\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(x2 + 1) dy + (2y − 1) dx = 0
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0 "given that" "y" = 0 "when" "x" = 1`.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
tan–1x + tan–1y = c is the general solution of the differential equation ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
