Advertisements
Advertisements
प्रश्न
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)`.
Advertisements
उत्तर
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
संबंधित प्रश्न
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
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
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Solve the differential equation `dy/dx -y =e^x`
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:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation (x2 + 1) \[\frac{dy}{dx}\] + (y2 + 1) = 0, is
The number of arbitrary constants in the particular solution of a differential equation of third order is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
\[\frac{dy}{dx} + y = 4x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \sqrt{4 - y^2}, - 2 < y < 2\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:- \[\left( x - y \right)\frac{dy}{dx} = x + 2y\]
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \frac{y}{x} = x^2\]
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Solve:
`2(y + 3) - xy (dy)/(dx)` = 0, given that y(1) = – 2.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of differential equation coty dx = xdy is ______.
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
