Advertisements
Advertisements
प्रश्न
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Advertisements
उत्तर
The given differential equation is dy = cosx(2 – y cosecx) dx
⇒ `"dy"/"dx"` = cosx(2 – y cosec x)
⇒ `"dy"/"dx"` = 2cosx – ycosx . cosecx
⇒ `"dy"/"dx"` = 2cosx – ycotx
⇒ `"dy"/"dx" + y cot x` = 2cosx
Here, P = cotx and Q = 2cosx.
∴ Integrating factor I.F. = `"e"^(intPdx)`
= `"e"^(int cot xdx)`
= `"e"^(log sinx)`
= sin x
∴ Required solution is `y xx "I"."F" = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y . sin x = int 2 cos x . sin x "d"x + "c"`
⇒ `y . sin x = int sin 2x "d"x + "c"`
⇒ `y . sin x = - 1/2 cos 2x + "c"`
Put x = `pi/2` and y = 2, we get
`2 sin pi/2 = - 1/2 cos pi + "c"`
⇒ 2(1) = `- 1/2 (-1) + "c"`
⇒ 2 = `1/2 + "c"`
⇒ c = `2 - 1/2 = 3/2`
∴ The equation is y sin x = `- 1/2 cos 2x + 3/2`.
APPEARS IN
संबंधित प्रश्न
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
The solution of the differential equation dy/dx = sec x – y tan x is:
(A) y sec x = tan x + c
(B) y sec x + tan x = c
(C) sec x = y tan x + c
(D) sec x + y tan x = c
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Solve the differential equation `cos^2 x dy/dx` + y = tan x
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
Which of the following differential equations has y = x as one of its particular solution?
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is
\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.
The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
