Advertisements
Advertisements
प्रश्न
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
विकल्प
sin x
sec x
tan x
cos x
Advertisements
उत्तर
sec x
We have,
\[\cos x\frac{dy}{dx} + y \sin x = 1\]
Dividing both sides by cos x, we get
\[\frac{dy}{dx} + \frac{\sin x}{\cos x}y = \frac{1}{\cos x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \tan x \right)y = \frac{1}{\cos x}\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = \tan x\]
\[Q = \frac{1}{\cos x}\]
Now,
\[ I . F . = e^{\int\tan xdx} = e^{log\left( \sec x \right)} \]
\[ = \sec x\]
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Verify that y = cx + 2c2 is a solution of the differential equation
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
(1 + x2) dy = xy dx
x cos2 y dx = y cos2 x dy
tan y dx + sec2 y tan x dy = 0
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
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 \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
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.
y2 dx + (xy + x2 ) dy = 0
x2y dx – (x3 + y3) dy = 0
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
Solve the differential equation
`y (dy)/(dx) + x` = 0
