Advertisements
Advertisements
Question
If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.
Solution
Given equation is `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx
⇒ `((2 + sin y)/(cos x))"dy"/"dx"` = –(1 + y)
⇒ `"dy"/((1 + y)) = -((cosx)/(2 + sinx))"d"x`
Integrating both sides, we get
`int "dy"/(1 + y) = - int cosx/(2 + sinx) "d"x`
⇒ `log|1 + y| = - log|2 + sinx| + logc`
⇒ `log|1 + y| + log|2 + sinx|` = log c
⇒ `log(1 + y)(2 + sinx)` = log c
⇒ `(1 + y)(2 + sinx)` = c
Put x = 0 and y = 1, we get
(1 + 1)(2 + sin 0) = c
⇒ 4 = c
∴ Equation is (1 + y)(2 + sinx) = 4
Now put x = `pi/2`
∴ `(1 + y)(2 + sin pi/2)` = 4
⇒ (1 + y)(2 + 1) = 4
⇒ 1 + y = `4/3`
⇒ y = `4/3 - 1`
⇒ `1/3`
So, `y(pi/2) = 1/3`
Hence, the required solution is `y(pi/2) = 1/3`.
APPEARS IN
RELATED QUESTIONS
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Find the particular solution of the differential equation x (1 + y2) dx – y (1 + x2) dy = 0, given that y = 1 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The number of arbitrary constants in the particular solution of a differential equation of third order is
Which of the following differential equations has y = x as one of its particular solution?
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
cos (x + y) dy = dx
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
x2 dy + (x2 − xy + y2) dx = 0
\[\cos^2 x\frac{dy}{dx} + y = \tan x\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
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`
Solve the differential equation:
`(xdy - ydx) ysin(y/x) = (ydx + xdy) xcos(y/x)`.
Find the particular solution satisfying the condition that y = π when x = 1.
