Advertisements
Advertisements
प्रश्न
For the differential equation given, find a particular solution satisfying the given condition:
`dy/dx - 3ycotx = sin 2x; y = 2` when `x = pi/2`
Advertisements
उत्तर
The given equation is
`dy/dx - 3 y cot x = sin 2x` ....(1)
Which is a linear equation of the type
`dy/dx + Py = Q`
Here P = - 3cot x and Q = sin 2x
∴ `intP dx = -3 int cot x dx = -3 log |sin x|`
∴ `I.F. = e^(-3log|sin x|)`
`= e^(log cosec^3 x)`
`= cosec^3 x`
∴ The solution is `y. (I.F.) = int Q. (I.F.) dx + C`
`y cosec^3 x = int sin2x cosec^3x dx + C`
`= int (2 sin x cos x)/(sin^3 x) dx + C`
`= 2 int cosec x cot x dx + C`
`= - 2 cosec x +C`
⇒ y = -2 sin2 x + C sin3 x ....(2)
When `x = pi/2, y = 2`
∴ `2 = -2 sin^2 pi/2 + C sin^3 pi/2`
⇒ 2 = -2 (1)2 + C (1)3
⇒ C = 2 + 2
⇒ C = 4
Putting in (2), we get
y = - 2sin2 x + 4 sin3 x
⇒ y = 4 sin3 x - 2 sin x
Which is the required solution.
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`x dy/dx + 2y= x^2 log x`
For the differential equation, find the general solution:
`x log x dy/dx + y= 2/x log x`
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 0)`
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 Integrating Factor of the differential equation `dy/dx - y = 2x^2` is ______.
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?
\[\frac{dy}{dx}\] = y tan x − 2 sin x
Find the general solution of the differential equation \[x\frac{dy}{dx} + 2y = x^2\]
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 \[y = \frac{\pi}{2}\].
Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .
Find the integerating factor of the differential equation `x(dy)/(dx) - 2y = 2x^2`
Solve the following differential equation:
`("x + y") "dy"/"dx" = 1`
Solve the following differential equation:
dr + (2r cotθ + sin2θ)dθ = 0
Solve the following differential equation:
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
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.
Integrating factor of `dy/dx + y = x^2 + 5` is ______
Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 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.
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 ______.
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 ______.
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.
If sec x + tan x is the integrating factor of `dy/dx + Py` = Q, then value of P is ______.
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 ______.
