Advertisements
Advertisements
प्रश्न
For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]
Advertisements
उत्तर
We have,
\[\cos \left( \frac{dy}{dx} \right) = a\]
\[ \Rightarrow \frac{dy}{dx} = \cos^{- 1} a\]
\[ \Rightarrow dy = \cos^{- 1} a dx\]
Integrating both sides, we get
\[\int dy = \int \cos^{- 1} a dx\]
\[ \Rightarrow y = x \cos^{- 1} a + C\]
Now,
When `x = 0, y = 1`
\[ \therefore 1 = 0 + C\]
\[ \Rightarrow C = 1\]
Putting the value of `C` in (1), we get
\[y = x \cos^{- 1} a + 1\]
\[ \Rightarrow \cos\left( \frac{y - 1}{x} \right) = a\]
APPEARS IN
संबंधित प्रश्न
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`
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 `(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.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.
Solve the differential equation `cos^2 x dy/dx` + y = tan x
The solution of the differential equation \[\frac{dy}{dx} = 1 + x + y^2 + x y^2 , y\left( 0 \right) = 0\] is
The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents
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
The solution of x2 + y2 \[\frac{dy}{dx}\]= 4, is
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
Write the solution of the differential equation \[\frac{dy}{dx} = 2^{- y}\] .
`y sec^2 x + (y + 7) tan x(dy)/(dx)=0`
\[\frac{dy}{dx} + 2y = \sin 3x\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \left( 1 + x^2 \right)\left( 1 + y^2 \right)\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} + y = 1\]
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:-
\[x\frac{dy}{dx} + 2y = x^2 , x \neq 0\]
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 general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
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)`.
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`.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
