Advertisements
Advertisements
प्रश्न
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Advertisements
उत्तर
`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
संबंधित प्रश्न
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 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 = cos x + C : y′ + sin x = 0
Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
If m and n are the order and degree of the differential equation \[\left( y_2 \right)^5 + \frac{4 \left( y_2 \right)^3}{y_3} + y_3 = x^2 - 1\], then
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
(x + y − 1) dy = (x + y) dx
(x3 − 2y3) dx + 3x2 y dy = 0
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:-
\[\frac{dy}{dx} + 2y = \sin x\]
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Find a particular solution of the following differential equation:- (x + y) dy + (x − y) dx = 0; y = 1 when x = 1
Find the differential equation of all non-horizontal lines in a plane.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 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 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 ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The value of c in the particular solution given that y(0) = 0 and k = 0.049 is ______.
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
The differential equation of all parabolas that have origin as vertex and y-axis as axis of symmetry is ______.
