Karnataka Board PUCPUC Science 2nd PUC Class 12

Find the Particular Solution of the Differential Equation D X D Y + X Cot Y = 2 Y + Y 2 Cot Y , Y ≠ 0 Given that X = 0 When Y = π 2 . - Mathematics

Advertisements
Advertisements
Sum

Find the particular solution of the differential equation \[\frac{dx}{dy} + x \cot y = 2y + y^2 \cot y, y ≠ 0\] given that x = 0 when \[y = \frac{\pi}{2}\].

Advertisements

Solution

We have,
\[\frac{dx}{dy} + x \cot y = 2y + y^2 \cot y . . . . . \left( 1 \right)\]
Clearly, it is a linear differential equation of the form
\[\frac{dx}{dy} + Px = Q\]
\[\text{ where }P = \cot y\text{ and }Q = 2y + y^2 \cot y\]
\[ \therefore I . F . = e^{\int P\ dy} \]
\[ = e^{\int\cot y\ dy} \]
\[ = e^{log\left| \sin y \right|} = \sin y\]
\[\text{Multiplying both sides of } \left( 1 \right)\text{ by }I.F. = \sin y,\text{ we get }\]
\[\sin y\left( \frac{dx}{dy} + x \cot y \right) = \sin y\left( y^2 \cot y + 2y \right)\]
\[ \Rightarrow \sin y\frac{dx}{dy} + x \cos y = y^2 \cos y + 2y \sin y\]
Integrating both sides with respect to y, we get

\[ \Rightarrow x \sin y = y^2 \int\cos y\ dy - \int\left[ \frac{d}{dy}\left( y^2 \right)\int\cos y\ dy \right]dy + \int2y \sin y\ dy + C\]
\[ \Rightarrow x \sin y = y^2 \sin y - \int2y \sin y\ dy + \int2y\sin y\ dy + C\]
\[ \Rightarrow x \sin y = y^2 \sin y + C\]
Now, 
\[y = \frac{\pi}{2}\text{ at }x = 0\]
\[ \therefore 0 \times \sin \frac{\pi}{2} = \frac{\pi}{4}^2 \sin \frac{\pi}{2} + C\]
\[ \Rightarrow C = - \frac{\pi}{4}^2 \]
Putting the value of C, we get
\[x \sin y = y^2 \sin y - \frac{\pi}{4}^2 \]
\[\text{ Hence, }x \sin y = y^2 \sin y - \frac{\pi}{4}^2\text{ is the required solution.}\]

  Is there an error in this question or solution?
Chapter 22: Differential Equations - Exercise 22.10 [Page 108]

APPEARS IN

RD Sharma Class 12 Maths
Chapter 22 Differential Equations
Exercise 22.10 | Q 41 | Page 108

RELATED QUESTIONS

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.


For the differential equations find the general solution:

`dy/dx  + 2y = sin x`


For the differential equations find the general solution:

`dy/dx + secxy = tan x (0 <= x < pi/2)`


For the differential equations find the general solution:

`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`


For the differential equations find the general solution:

`x dy/dx +  2y= x^2 log x`


For the differential equations find the general solution:

`x dy/dx + y - x + xycot x = 0(x != 0)`


For the differential equations find the general solution:

y dx + (x – y2) dy = 0

 


For the differential equations given find a particular solution satisfying the given condition:

`(1 + x^2)dy/dx + 2xy = 1/(1 + x^2); y = 0 " when x " =1`


For the differential equations given find a particular solution satisfying the given condition:

`dy/dx - 3ycotx = sin 2x; y = 2 " when x  "= pi/2`


Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.


Find the equation of a 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 population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?


Solve the differential equation `(tan^(-1) x- y) dx = (1 + x^2) dy`


Find the general solution of the differential equation `dy/dx - y = sin x`


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

\[y^2 \frac{dx}{dy} + x - \frac{1}{y} = 0\]

 


(x + tan y) dy = sin 2y dx


dx + xdy = e−y sec2 y dy


\[\frac{dy}{dx}\] = y tan x − 2 sin x


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


\[\left( \sin x \right)\frac{dy}{dx} + y \cos x = 2 \sin^2 x \cos x\]

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

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

Solve the differential equation \[\left( x + 2 y^2 \right)\frac{dy}{dx} = y\], given that when x = 2, y = 1.


Find the general solution of the differential equation \[x\frac{dy}{dx} + 2y = x^2\]


Find the general solution of the differential equation \[\frac{dy}{dx} - y = \cos x\]


Solve the following differential equation:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]


Solve the differential equation: `(1 + x^2) dy/dx + 2xy - 4x^2 = 0,` subject to the initial condition y(0) = 0.


Solve the following differential equation:

`"dy"/"dx" + "y"/"x" = "x"^3 - 3`


Solve the following differential equation:

`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`


Solve the following differential equation:

`("x" + 2"y"^3) "dy"/"dx" = "y"`


Solve the following differential equation:

`"dy"/"dx" + "y" * sec "x" = tan "x"`


Solve the following differential equation:

`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`


Solve the following differential equation:

`("x + y") "dy"/"dx" = 1`


Solve the following differential equation:

`("x + a")"dy"/"dx" - 3"y" = ("x + a")^5`


Solve the following differential equation:

dr + (2r cot θ + sin 2θ) dθ = 0


Solve the following differential equation:

y dx + (x - y2) dy = 0


Solve the following differential equation:

`(1 - "x"^2) "dy"/"dx" + "2xy" = "x"(1 - "x"^2)^(1/2)`


Solve the following differential equation:

`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`


Find the equation of the curve which passes through the origin and has the slope x + 3y - 1 at any point (x, y) on it.


Find the equation of the curve passing through the point `(3/sqrt2, sqrt2)` having a slope of the tangent to the curve at any point (x, y) is -`"4x"/"9y"`.


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


If the slope of the tangent to the curve at each of its point is equal to the sum of abscissa and the product of the abscissa and ordinate of the point. Also, the curve passes through the point (0, 1). Find the equation of the curve.


Form the differential equation of all circles which pass through the origin and whose centers lie on X-axis.


The integrating factor of `(dy)/(dx) + y` = e–x is ______.


`(x + 2y^3 ) dy/dx = y`


Find the general solution of the equation `("d"y)/("d"x) - y` = 2x.

Solution: The equation `("d"y)/("d"x) - y` = 2x

is of the form `("d"y)/("d"x) + "P"y` = Q

where P = `square` and Q = `square`

∴ I.F. = `"e"^(int-"d"x)` = e–x

∴ the solution of the linear differential equation is

ye–x = `int 2x*"e"^-x  "d"x + "c"`

∴ ye–x  = `2int x*"e"^-x  "d"x + "c"`

= `2{x int"e"^-x "d"x - int square  "d"x* "d"/("d"x) square"d"x} + "c"`

= `2{x ("e"^-x)/(-1) - int ("e"^-x)/(-1)*1"d"x} + "c"`

∴ ye–x = `-2x*"e"^-x + 2int"e"^-x "d"x + "c"`

∴ e–xy = `-2x*"e"^-x+ 2 square + "c"`

∴ `y + square + square` = cex is the required general solution of the given differential equation


The integrating factor of the differential equation sin y `("dy"/"dx")` = cos y(1 - x cos y) is ______.


The integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.


The slope of the tangent to the curves x = 4t3 + 5, y = t2 - 3 at t = 1 is ______


Integrating factor of `dy/dx + y = x^2 + 5` is ______ 


The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.


The equation x2 + yx2 + x + y = 0 represents


The integrating factor of the differential equation `x (dy)/(dx) - y = 2x^2` is


The integrating factor of differential equation `(1 - y)^2  (dx)/(dy) + yx = ay(-1 < y < 1)`


State whether the following statement is true or false.

The integrating factor of the differential equation `(dy)/(dx) + y/x` = x3 is – x.


Let y = y(x), x > 1, be the solution of the differential equation `(x - 1)(dy)/(dx) + 2xy = 1/(x - 1)`, with y(2) = `(1 + e^4)/(2e^4)`. If y(3) = `(e^α + 1)/(βe^α)`, then the value of α + β is equal to ______.


If y = y(x) is the solution of the differential equation, `(dy)/(dx) + 2ytanx = sinx, y(π/3)` = 0, then the maximum value of the function y (x) over R is equal to ______.


Let y = y(x) be a solution curve of the differential equation (y + 1)tan2xdx + tanxdy + ydx = 0, `x∈(0, π/2)`. If `lim_(x→0^+)` xy(x) = 1, then the value of `y(π/2)` is ______.


Let y = f(x) be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If f(x) satisfies xf'(x) = x2 + f(x) – 2, then the area bounded by f(x) with x-axis between ordinates x = 0 and x = 3 is equal to ______.


Let y = y(x) be the solution curve of the differential equation `(dy)/(dx) + ((2x^2 + 11x + 13)/(x^3 + 6x^2 + 11x + 6)) y = ((x + 3))/(x + 1), x > - 1`, which passes through the point (0, 1). Then y(1) is equal to ______.


Let the solution curve y = y(x) of the differential equation (4 + x2) dy – 2x (x2 + 3y + 4) dx = 0 pass through the origin. Then y (2) is equal to ______.


If the solution curve y = y(x) of the differential equation y2dx + (x2 – xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = `sqrt(3)  x` at the point `(α, sqrt(3) α)`, then value of `log_e (sqrt(3)α)` is equal to ______.


If the slope of the tangent at (x, y) to a curve passing through `(1, π/4)` is given by `y/x - cos^2(y/x)`, then the equation of the curve is ______.


If sin x is the integrating factor (IF) of the linear differential equation `dy/dx + Py` = Q then P is ______.


The solution of the differential equation `dx/dt = (xlogx)/t` is ______.


Find the general solution of the differential equation:

`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`


Solve the differential equation `dy/dx+2xy=x` by completing the following activity.

Solution: `dy/dx+2xy=x`       ...(1)

This is the linear differential equation of the form `dy/dx +Py =Q,"where"`

`P=square` and Q = x

∴ `I.F. = e^(intPdx)=square`

The solution of (1) is given by

`y.(I.F.)=intQ(I.F.)dx+c=intsquare  dx+c`

∴ `ye^(x^2) = square`

This is the general solution.


The slope of tangent at any point on the curve is 3. lf the curve passes through (1, 1), then the equation of curve is ______.


Solve:

`xsinx dy/dx + (xcosx + sinx)y` = sin x


The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.


Share
Notifications



      Forgot password?
Use app×