Advertisements
Advertisements
Question
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Advertisements
Solution
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
RELATED QUESTIONS
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
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
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
Find the particular solution of the differential equation `e^xsqrt(1-y^2)dx+y/xdy=0` , given that y=1 when x=0
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 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.
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 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
if `y = sin^(-1) (6xsqrt(1-9x^2))`, `1/(3sqrt2) < x < 1/(3sqrt2)` then find `(dy)/(dx)`
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
Write the order of the differential equation associated with the primitive y = C1 + C2 ex + C3 e−2x + C4, where C1, C2, C3, C4 are arbitrary constants.
The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if
The number of arbitrary constants in the particular solution of a differential equation of third order is
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} + y = 4x\]
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
For the following differential equation, find a particular solution satisfying the given condition:
\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the following differential equation:-
\[x \log x\frac{dy}{dx} + y = \frac{2}{x}\log x\]
Solve the following differential equation:-
\[\left( x + y \right)\frac{dy}{dx} = 1\]
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solution of differential equation xdy – ydx = 0 represents : ______.
y = aemx+ be–mx satisfies which of the following differential equation?
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
