Advertisements
Advertisements
प्रश्न
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
पर्याय
cos x
tan x
sec x
sin x
उत्तर
sec x
We have,
\[\cos x\frac{dy}{dx} + y \sin x = 1\]
Dividing both sides by cos x, we get
\[\frac{dy}{dx} + \frac{\sin x}{\cos x}y = \frac{1}{\cos x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \tan x \right)y = \frac{1}{\cos x}\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = \tan x\]
\[Q = \frac{2}{\cos x}\]
Now,
\[I . F . = e^{\int\tan xdx} \]
\[ = e^{log\left( sec x \right)} \]
\[ = \sec x\]
APPEARS IN
संबंधित प्रश्न
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
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 \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
(sin x + cos x) dy + (cos x − sin x) dx = 0
x cos2 y dx = y cos2 x dy
xy dy = (y − 1) (x + 1) dx
dy + (x + 1) (y + 1) dx = 0
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
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 decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
y2 dx + (xy + x2)dy = 0
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
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
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
