Advertisements
Advertisements
प्रश्न
(1 + x) y dx + (1 + y) x dy = 0
उत्तर
We have,
(1 + x) y dx + (1 + y) x dy = 0
\[\frac{dy}{dx} = - \frac{y\left( 1 + x \right)}{x\left( 1 + y \right)}\]
\[ \Rightarrow \left( \frac{1 + y}{y} \right)dy = - \left( \frac{1 + x}{x} \right)dx\]
\[ \Rightarrow \left( \frac{1}{y} + y \right)dy = - \left( \frac{1}{x} + 1 \right)dx\]
Integrating both sides, we get
\[\int\left( \frac{1}{y} + 1 \right)dy = - \int\left( \frac{1}{x} + 1 \right)dx\]
\[ \Rightarrow \int\frac{1}{y}dy + \int dy = - \int\frac{1}{x}dx - \int dx\]
\[ \Rightarrow \log\left| y \right| + y = - \log\left| x \right| - x + C\]
\[ \Rightarrow \log\left| xy \right| + y + x = C\]
\[ \Rightarrow x + y + \log\left| xy \right| = C\]
APPEARS IN
संबंधित प्रश्न
Write the integrating factor of the following differential equation:
(1+y2) dx−(tan−1 y−x) dy=0
Find the differential equation of the family of lines passing through the origin.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
`x/a + y/b = 1`
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e– 2x
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = ex (a cos x + b sin x)
Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`
Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)
The general solution of a differential equation of the type `dx/dy + P_1 x = Q_1` is ______.
Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.
Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Find the differential equation corresponding to y = ae2x + be−3x + cex where a, b, c are arbitrary constants.
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
\[\frac{dy}{dx} = y^2 + 2y + 2\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]
tan y dx + tan x dy = 0
x cos2 y dx = y cos2 x dy
cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy
cosec x (log y) dy + x2y dx = 0
Find the general solution of the differential equation `"dy"/"dx" = y/x`.
A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.
Solve the differential equation:
cosec3 x dy − cosec y dx = 0
The general solution of the differential equation `(dy)/(dx) = e^(x + y)` is
The general solution of the differential equation of the type `(dx)/(dy) + p_1y = theta_1` is
The general solution of the differential equation `x^xdy + (ye^x + 2x) dx` = 0
Find the general solution of differential equation `(dy)/(dx) = (1 - cosx)/(1 + cosx)`
The general solution of the differential equation y dx – x dy = 0 is ______.
The general solution of the differential equation ydx – xdy = 0; (Given x, y > 0), is of the form
(Where 'c' is an arbitrary positive constant of integration)
