Advertisements
Advertisements
Question
Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.
`(dy)/(dx) + ycotx = 2/(1 + sinx)`
Advertisements
Solution
The differential equation is a linear differential equation
IF = `e^(int cotxdx) = e^(logsinx) = sinx`
The general solution is given by
`ysinx = int 2 sinx/(1 + sinx) dx`
⇒ `ysinx = 2 int (sinx + 1 - 1)/(1 + sinx) dx = 2 int [1 - 1/(1 + sinx)] dx`
⇒ `ysinx = 2 int [1 - 1/(1 + cos(pi/2 - x))] dx`
⇒ `ysinx = 2 int [1 - 1/(2cos^2 (pi/4 - x/2))] dx`
⇒ `ysinx = 2 int [1 - 1/2 sec^2 (pi/4 - x/2)] dx`
⇒ `ysinx = 2[x + tan(pi/4 - x/2)] + c`
Given that y = 0, when x = `pi/4`,
Hence, 0 = `2[pi/4 + tan pi/8] + c`
⇒ `c = - pi/2 - 2 tan pi/8`
Hence, the particular solution is `y = "cosec"x [2{x + tan (pi/4 - x/2)} - (pi/2 + 2tan pi/8)]`
APPEARS IN
RELATED QUESTIONS
Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0
Also, find the particular solution when x = 0 and y = π.
Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
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)`
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
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.
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The number of arbitrary constants in the particular solution of a differential equation of third order is
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + y = 4x\]
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`
Find the general solution of `"dy"/"dx" + "a"y` = emx
If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
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 ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
The solution of differential equation coty dx = xdy is ______.
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
