Advertisements
Advertisements
प्रश्न
`x cos x(dy)/(dx)+y(x sin x + cos x)=1`
Advertisements
उत्तर
We have,
\[x \cos x\frac{dy}{dx} + y \left( x \sin x + \cos x \right) = 1\]
\[ \Rightarrow \frac{dy}{dx} + \left( \tan x + \frac{1}{x} \right)y = \frac{1}{x \cos x}\]
\[\text{Comparing with }\frac{dy}{dx} + Px = Q,\text{ we get}\]
\[P = \tan x + \frac{1}{x} \]
\[Q = \frac{1}{x \cos x}\]
Now,
\[I . F . = e^{\int\left( \tan x + \frac{1}{x} \right) dx} = e^{\log \left| x \sec x \right|} = x \sec x\]
So, the solution is given by
\[xy \sec x = \int \sec^2 x dx + C\]
\[ \Rightarrow xy \sec x = \tan x + C\]
APPEARS IN
संबंधित प्रश्न
The differential equation of the family of curves y=c1ex+c2e-x is......
(a)`(d^2y)/dx^2+y=0`
(b)`(d^2y)/dx^2-y=0`
(c)`(d^2y)/dx^2+1=0`
(d)`(d^2y)/dx^2-1=0`
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y″ – y′ = 0
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`
Find the particular solution of the differential equation
`tan x * (dy)/(dx) = 2x tan x + x^2 - y`; `(tan x != 0)` given that y = 0 when `x = pi/2`
The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is
The general solution of the differential equation \[\frac{dy}{dx} + y \] cot x = cosec x, is
The solution of the differential equation \[\frac{dy}{dx} + \frac{2y}{x} = 0\] with y(1) = 1 is given by
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 \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is
The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is
The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
x (e2y − 1) dy + (x2 − 1) ey dx = 0
\[\frac{dy}{dx} + 1 = e^{x + y}\]
\[\frac{dy}{dx} - y \tan x = e^x\]
(1 + y + x2 y) dx + (x + x3) dy = 0
x2 dy + (x2 − xy + y2) dx = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[\frac{dy}{dx} + 5y = \cos 4x\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
\[y^2 + \left( x + \frac{1}{y} \right)\frac{dy}{dx} = 0\]
For the following differential equation, find the general solution:- \[\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Find the equation of the curve passing through the point (1, 1) whose differential equation is x dy = (2x2 + 1) dx, x ≠ 0.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x.\]
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
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)`.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Solution of differential equation xdy – ydx = 0 represents : ______.
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
The general solution of ex cosy dx – ex siny dy = 0 is ______.
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.
The differential equation for which y = acosx + bsinx is a solution, is ______.
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
The member of arbitrary constants in the particulars solution of a differential equation of third order as
