Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
Advertisements
उत्तर
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
∴ `"dy"/"dx" + 1/(1 + "x"^2) * "y" = "e"^(tan^-1 "x")/(1 + "x"^2)` ....(1)
This is the linear differential equation of the form
`"dy"/"dx" + "P" * "y" = "Q",` where P = `1/(1 + "x"^2)` and Q = `"e"^(tan^-1 "x")/(1 + "x"^2)`
∴ I.F. = `"e"^(int "P dx") = "e"^(int 1/"1 + x"^2"dx")`
`= "e"^(tan^-1 "x")`
∴ the solution of (1) is given by
`"y" * ("I.F.") = int "Q" * ("I.F.") "dx" + "c"`
∴ `"y" * "e"^(tan^-1"x") = int "e"^(tan^-1 "x")/(1 + "x"^2) * "e"^(tan^-1"x") "dx" + "c"`
∴ `"y" * "e"^(tan^-1"x") = int ("e"^(tan^-1 "x")) * ("e"^(tan^-1"x")/(1 + "x"^2)) "dx" + "c"`
Put `"e"^(tan^-1"x") = "t"`
∴ `"e"^(tan^-1"x")/(1 + "x"^2) "dx" = "dt"`
∴ `"y" * "e"^(tan^-1"x") = int "t dt" + "c"`
∴ `"y" * "e"^(tan^-1"x") = "t"^2/2 + "c"`
∴ `"y" * "e"^(tan^-1"x") = 1/2 ("e"^(tan^-1"x"))^2 + "c"`
∴ y = `1/2 "e"^(tan^-1"x") + "ce"^(- tan^-1 "x")`
This is the general solution.
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`dy/dx + 2y = sin x`
For the differential equation, find the general solution:
`dy/dx + 3y = e^(-2x)`
For the differential equation, find the general solution:
`dy/dx + y/x = x^2`
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:
(1 + x2) dy + 2xy dx = cot x dx (x ≠ 0)
For the differential equation, find the general solution:
`(x + y) dy/dx = 1`
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 0)`
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 integrating factor of the differential equation.
`(1 - y^2) dx/dy + yx = ay(-1 < y < 1)` 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?
x dy = (2y + 2x4 + x2) dx
\[\frac{dy}{dx}\] = y tan x − 2 sin x
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:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]
If f(x) = x + 1, find `"d"/"dx"("fof") ("x")`
Solve the following differential equation:
`("x + a")"dy"/"dx" - 3"y" = ("x + a")^5`
Solve the following differential equation:
`(1 - "x"^2) "dy"/"dx" + "2xy" = "x"(1 - "x"^2)^(1/2)`
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 integrating factor of `(dy)/(dx) + y` = e–x is ______.
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 solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.
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 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
