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
Solve the differential equation: `x+ydy/dx=sec(x^2+y^2)` Also find the particular solution if x = y = 0.
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`
How many arbitrary constants are there in the general solution of the differential equation of order 3.
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation x dx + y dy = x2 y dy − y2 x dx, is
The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], 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 ex dy + (y ex + 2x) dx = 0 is
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} = \left( x + y \right)^2\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:- `y dx + x log (y)/(x)dy-2x dy=0`
Solve the following differential equation:-
(1 + x2) dy + 2xy dx = cot x dx
Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
General solution of `("d"y)/("d"x) + ytanx = secx` is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
Find a particular solution satisfying the given condition `- cos((dy)/(dx)) = a, (a ∈ R), y` = 1 when `x` = 0
Find a particular solution, satisfying the condition `(dy)/(dx) = y tan x ; y = 1` when `x = 0`
Find the general solution of the differential equation:
`log((dy)/(dx)) = ax + by`.
