Advertisements
Advertisements
प्रश्न
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Advertisements
उत्तर
sec2y tan x dy + sec2x tan y dx = 0
Dividing both sides by tan x tan y, we get
`(sec^2y tan x)/(tanx tan y) "d"y + (sec^2x tany)/(tanx tany) "d"x` = 0
∴ `(sec^2x)/(tanx) "d"x + (sec^2y)/(tany) "d"y` = 0
Integrating on both sides, we get
`int (sec^2x)/(tanx) "d"x + int (sec^2y)/(tany) "d"y` = 0
∴ log |tan x| + log |tan y| = log |c|
∴ log |tan x.tan y| = log |c|
∴ tan x tan y = c
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
C' (x) = 2 + 0.15 x ; C(0) = 100
xy (y + 1) dy = (x2 + 1) dx
dy + (x + 1) (y + 1) dx = 0
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The differential equation satisfied by ax2 + by2 = 1 is
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
y2 dx + (x2 − xy + y2) dy = 0
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
The price of six different commodities for years 2009 and year 2011 are as follows:
| Commodities | A | B | C | D | E | F |
|
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
| Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
`dy/dx + y` = 3
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Solve
`dy/dx + 2/ x y = x^2`
y2 dx + (xy + x2)dy = 0
x2y dx – (x3 + y3) dy = 0
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
