Solve the Differential Equation : X D Y D X + Y − X + Xy Cot X = 0 ; X ≠ 0 . - Mathematics

Advertisements
Advertisements
Sum

Solve the differential equation : `"x"(d"y")/(d"x") + "y" - "x" + "xy"cot"x" = 0; "x" != 0.`

Advertisements

Solution

`x dy/dx + y - x + xy cos x = 0`

⇒ `dy/dx + (1/x + cot x ) y = 1`

I.F. = `e^(int(1/x + cot x) dx) = e^(log(x sin x))`

= x sin x

`∴ y xx  x sin x = x sinx dx`

⇒ xy sin x = - x cos x + sin x + C

  Is there an error in this question or solution?
2015-2016 (March) All India Set 1 E

RELATED QUESTIONS

Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`


Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`


Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`


Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`


Solve `sin x dy/dx - y = sin x.tan  x/2`


Solve the differential equation `sin^(-1) (dy/dx) = x + y`


\[\frac{dy}{dx} + 2y = e^{3x}\]

\[4\frac{dy}{dx} + 8y = 5 e^{- 3x}\]

\[\frac{dy}{dx} + 2y = 6 e^x\]

\[\frac{dy}{dx} + y = e^{- 2x}\]

\[x\frac{dy}{dx} = x + y\]

\[\frac{dy}{dx} + 2y = 4x\]

\[\frac{dy}{dx} + \frac{4x}{x^2 + 1}y + \frac{1}{\left( x^2 + 1 \right)^2} = 0\]

\[x\frac{dy}{dx} + y = x \log x\]

\[\frac{dy}{dx} + \frac{y}{x} = x^3\]

\[\frac{dy}{dx} + 2y = \sin x\]

\[\frac{dy}{dx}\] = y tan x − 2 sin x


\[\frac{dy}{dx}\] + y tan x = cos x


\[\left( 1 + y^2 \right) + \left( x - e^{tan^{- 1} y} \right)\frac{dy}{dx} = 0\]

Find the equation of the curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.


The decay rate of radium at any time  t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.


Solve the differential equation: (x + 1) dy – 2xy dx = 0


Solve the differential equation: (1 + x2) dy + 2xy dx = cot x dx


Solve the differential equation `"dy"/"dx" + y/x` = x2.


`"dy"/"dx" + y` = 5 is a differential equation of the type `"dy"/"dx" + "P"y` = Q but it can be solved using variable separable method also.


`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.


Solution of the differential equation of the type `("d"x)/("d"y) + "p"_1x = "Q"_1` is given by x.I.F. = `("I"."F") xx "Q"_1"d"y`.


Correct substitution for the solution of the differential equation of the type `("d"y)/("d"x) = "f"(x, y)`, where f(x, y) is a homogeneous function of zero degree is y = vx.


If ex + ey = ex+y, then `"dy"/"dx"` 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 solution of the differential equation `"dy"/"dx" = "k"(50 - "y")` is given by ______.


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.

Which of the following solutions may be used to find the number of children who have been given the polio drops?


Solve the differential equation:

`"dy"/"dx" = 2^(-"y")`


The solution of the differential equation `(dx)/(dy) + Px = Q` where P and Q are constants or functions of y, is given by


If α, β are different values of x satisfying the equation a cos x + b sinα x = c, where a, b and c are constants, then `tan ((alpha + beta)/2)` is


The solution of the differential equation `(dy)/(dx) = 1 + x + y + xy` when y = 0 at x = – 1 is


`int cos(log x)  dx = F(x) + C` where C is arbitrary constant. Here F(x) =


If `x (dy)/(dx) = y(log y - log x + 1)`, then the solution of the dx equation is


Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.


Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`


Solve the following differential equation: (y – sin2x)dx + tanx dy = 0


Let y = y(x) be the solution of the differential equation `(dy)/(dx) + (sqrt(2)y)/(2cos^4x - cos2x) = xe^(tan^-1(sqrt(2)cost2x)), 0 < x < π/2` with `y(π/4) = π^2/32`. If `y(π/3) = π^2/18e^(-tan^-1(α))`, then the value of 3α2 is equal to ______.


If y = y(x) is the solution of the differential equation `(1 + e^(2x))(dy)/(dx) + 2(1 + y^2)e^x` = 0 and y(0) = 0, then `6(y^'(0) + (y(log_esqrt(3))))^2` is equal to ______.


The population P = P(t) at time 't' of a certain species follows the differential equation `("dp")/("dt")` = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is ______.


Let y = y(x) be the solution of the differential equation `xtan(y/x)dy = (ytan(y/x) - x)dx, -1 ≤ x ≤ 1, y(1/2) = π/6`. Then the area of the region bounded by the curves x = 0, x = `1/sqrt(2)` and y = y(x) in the upper half plane is ______.


Let y = y(x) be the solution of the differential equation `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1. Then, the value of (y(3))2 is equal to ______.


Let y = y(x) be the solution of the differential equation, `(2 + sinxdy)/(y + 1) (dy)/(dx)` = –cosx. If y > 0, y(0) = 1. If y(π) = a, and `(dy)/(dx)` at x = π is b, then the ordered pair (a, b) is equal to ______.


Let y = y(x) be the solution of the differential equation, `(x^2 + 1)^2 ("dy")/("d"x) + 2x(x^2 + 1)"y"` = 1, such that y(0) = 0. If `sqrt("ay")(1) = π/32` then the value of  'a' is ______.


If y = f(x), f'(0) = f(0) = 1 and if y = f(x) satisfies `(d^2y)/(dx^2) + (dy)/(dx)` = x, then the value of [f(1)] is ______ (where [.] denotes greatest integer function)


The solution of the differential equation `(1 + y^2) + (x - e^(tan^-1y)) (dy)/(dx)` = 0, is ______.


Solve the differential equation: 

`dy/dx` = cosec y


Share
Notifications



      Forgot password?
Use app×