Advertisements
Advertisements
प्रश्न
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.`
Advertisements
उत्तर
Given the differential equation
`dy/dx + y cot x = 4x cosec x` ....(1)
Comparing with the linear equation `dy/dx + Py = Q`,
When P = cot x, Q = 4x cosec x
∴ `I.F. = e^(int Pdx) = e^(int cot x dx) = e^(log |sin x|) = sin x`
∴ The solution is `y. (I.F.) = int Q. (I.F.) dx + C`
`therefore y sin x = int 4x cosec x sin x dx + C`
`= int 4x dx + C = + C`
`= (4x^2)/2 + C`
⇒ y sinx = 2x2 + C ....(2)
When `x = pi/2, y = 0`
∴ `0 = 2 (pi^2/4) + C`
⇒ `C = -pi^2/2`
Putting `C = pi^2/2` in (2),
`y sinx = 2x^2 - pi^2/2 ; (sin x ne 0)`
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
The differential equation of `y=c/x+c^2` is :
(a)`x^4(dy/dx)^2-xdy/dx=y`
(b)`(d^2y)/dx^2+xdy/dx+y=0`
(c)`x^3(dy/dx)^2+xdy/dx=y`
(d)`(d^2y)/dx^2+dy/dx-y=0`
If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`
Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`
Find the particular solution of the differential equation dy/dx=1 + x + y + xy, given that y = 0 when x = 1.
Solve the differential equation `dy/dx -y =e^x`
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
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)`
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
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that y = 0 when x = 0.
Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
The solution of the differential equation \[\frac{dy}{dx} + 1 = e^{x + y}\], is
The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\frac{dy}{dx} - y \tan x = e^x \sec x\]
(x2 + 1) dy + (2y − 1) dx = 0
\[y - x\frac{dy}{dx} = b\left( 1 + x^2 \frac{dy}{dx} \right)\]
\[x\frac{dy}{dx} + x \cos^2 \left( \frac{y}{x} \right) = y\]
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
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:- \[x \cos\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\]
Solve the following differential equation:-
\[x\frac{dy}{dx} + 2y = x^2 \log x\]
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find the equation of a curve passing through the point (−2, 3), given that the slope of the tangent to the curve at any point (x, y) is `(2x)/y^2.`
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
If y = e–x (Acosx + Bsinx), then y is a solution of ______.
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
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 number of arbitrary constants in the general solution of a differential equation of order three is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Find the general solution of the differential equation:
`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`
