Advertisements
Advertisements
Question
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Advertisements
Solution
`dy/dx=-(x+ycosx)/(1+sinx)`
⇒ `dy/dx+cosx/(1+sinx)y=x/(1+sinx )" ......i"`
This is a linear differential equation with
`P=cosx/(1+sinx),Q =-x/(1+sinx)`
`:.I.F. = e^intcosx/(1+sinx)dx`
= `e^log(1+sinx)`
= 1+ sinx
Multiplying both the sides of i by I.F. = 1 + sinx, we get
`(1+sinx)dy/dx+ycosx=-x`
Integrating with respect to x, we get
`y(1+sinx)=int-xdx+C`
`=>y =(2C-x^2)/(2(1+sinx)) " ....(ii)"`
Given that y = 1 when x = 0
`:.1=(2C)/(2(1+0))`
⇒ C =1 ................(iii)
Put iii in ii , we get
`y = (2-x^2)/(2(1+sinx))`
APPEARS IN
RELATED QUESTIONS
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x) y′ = y
The number of arbitrary constants in the particular solution of a differential equation of third 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.
How many arbitrary constants are there in the general solution of the differential equation of order 3.
Which of the following differential equations has y = x as one of its particular solution?
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
(1 + y + x2 y) dx + (x + x3) dy = 0
(x2 + 1) dy + (2y − 1) dx = 0
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
Solve the following differential equation:-
\[\frac{dy}{dx} - y = \cos x\]
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:-
y dx + (x − y2) dy = 0
Find a particular solution of the following differential equation:- x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of differential equation coty dx = xdy is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
