Advertisements
Advertisements
Question
(1 + x) y dx + (1 + y) x dy = 0
Advertisements
Solution
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
RELATED QUESTIONS
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
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.
y2 = a (b2 – x2)
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is ______.
Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.
Verify that xy = a ex + b e−x + x2 is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Show that y2 − x2 − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]
\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]
\[\frac{dy}{dx} = y^2 + 2y + 2\]
\[\frac{dy}{dx} + 4x = e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]
cosec x (log y) dy + x2y dx = 0
(1 − x2) dy + xy dx = xy2 dx
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 ______.
The general solution of the differential equation `(dy)/(dx) + x/y` = 0 is
General solution of tan 5θ = cot 2θ is
Which of the following equations has `y = c_1e^x + c_2e^-x` as the general solution?
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
What is the general solution of differential equation `(dy)/(dx) = sqrt(4 - y^2) (-2 < y < 2)`
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)
