Advertisements
Advertisements
Question
Solve the following differential equation:
`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`
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 + 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 + (sec x) y = tan x (0 <= x < pi/2)`
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 log x dy/dx + y= 2/x log x`
For the differential equation, find the general solution:
`(x + y) dy/dx = 1`
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 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?
Find the general solution of the differential equation `dy/dx - y = sin x`
x dy = (2y + 2x4 + x2) dx
(x + tan y) dy = sin 2y dx
\[\frac{dy}{dx}\] + y cos x = sin x cos 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 \[\frac{dy}{dx} - y = \cos x\]
Solve the differential equation \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
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 following differential equation:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]
Solve the following differential equation:
`"dy"/"dx" + "y"/"x" = "x"^3 - 3`
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" + "y" = "e"^(tan^-1 "x")`
The integrating factor of `(dy)/(dx) + y` = e–x is ______.
The integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.
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 = 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 ______.
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 ______.
