Advertisements
Advertisements
Question
Solve the following differential equation:
`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`
Advertisements
Solution
`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`
`∴ "dy"/"dx" + 1/cos^2"x" * "y" = (tan "x")/(cos^2"x")`
∴ `"dy"/"dx" + sec^2"x" * "y" = tan "x" * sec^2 "x"` ....(1)
This is the linear differential equation of the form
`"dy"/"dx" + "P"*"y" = "Q"`, where P = sec2x and Q = `tan "x" * sec^2 "x"`
∴ I.F. = `"e"^(int "Pdx") = "e"^(int sec^2"x" "dx") = "e"^(tan"x")`
∴ the solution of (1) is given by
`"y" * ("I.F.") = int "Q" (I.F.) "dx" + "c"`
∴ `"y"*"e"^"tan x" = int "tan x" * sec^2"x" * "e"^(tan"x") "dx" + "c"`
Put tan x = t
∴ `sec^2"x" "dx" = "dt"`
∴ `"y" * "e"^"tan x" = int "t" * "e"^"t" "dt" + "c"`
∴ `"y" * "e"^"tan x" = "t" int "e"^"t" "dt" - int["d"/"dt" ("t") int "e"^"t" "dt"] "dt" + "c"`
`= "t" * "e"^"t" - int 1 * "e"^"t" "dt" + "c"`
`= "t" * "e"^"t" - "e"^"t" + "c"`
`= "e"^"t" ("t - 1") + "c"`
∴ `"y" * "e"^"tan x" = "e"^"tan x" (tan"x" - 1) + "c"`
This is the general solution.
APPEARS IN
RELATED QUESTIONS
For the differential equation, find the general solution:
`dy/dx + 3y = e^(-2x)`
For the differential equation, find the general solution:
`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`
For the differential equation, find the general solution:
`x dy/dx + y - x + xy cot x = 0(x != 0)`
For the differential equation, find the general solution:
`(x + y) dy/dx = 1`
For the differential equation given, find a particular solution satisfying the given condition:
`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`
For the differential equation 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.
The Integrating Factor of the differential equation `dy/dx - y = 2x^2` is ______.
Find the general solution of the differential equation `dy/dx - y = sin x`
(x + tan y) dy = sin 2y dx
dx + xdy = e−y sec2 y dy
Find the general solution of the differential equation \[\frac{dy}{dx} - y = \cos x\]
Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
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: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .
Solve the differential equation \[\frac{dy}{dx}\] + y cot x = 2 cos x, given that y = 0 when x = \[\frac{\pi}{2}\] .
Solve the differential equation: (1 +x2 ) dy + 2xy dx = cot x dx
If f(x) = x + 1, find `"d"/"dx"("fof") ("x")`
Solve the following differential equation:
`dy/dx + y/x = x^3 - 3`
Solve the following differential equation:
`"dy"/"dx" + "y" * sec "x" = tan "x"`
Solve the following differential equation:
`("x + y") "dy"/"dx" = 1`
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)`
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 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 (1 + x2)dt = (tan-1 x - t)dx is ______.
Integrating factor of `dy/dx + y = x^2 + 5` is ______
Which of the following is a second order differential equation?
The equation x2 + yx2 + x + y = 0 represents
The integrating factor of differential equation `(1 - y)^2 (dx)/(dy) + yx = ay(-1 < y < 1)`
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 = 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 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 ______.
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 ______.
