Advertisements
Advertisements
Question
For the differential equation, find the general solution:
`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`
Solution
The given equation is
`cos^2 x dy/dx + y = tan x`
⇒ `dy/dx + (sec^2 x) y = tan x sec^2 x`
Which is a linear equation of the type
`dy/dx + Py = Q`
Here P = sec2 x and Q = tan sec2 x
∴ `I.F. = e^(intsec^2 x dx) = e^(tan x)`
∴ The solution is `y. (I.F.) = int Q. (I.F.) dx + C`
⇒ `y.e^(tan x) = int tan x sec^2 x e^(tan x) dx + C = I + C` ...(1)
Now, `I = int tan x sec^2 xe^(tan x) dx`
Put tan x = t
⇒ sec2 x dx = dt
∴ `I = int t. e^t dt = t. e^t - int (1) e^t dt` ....[Integrating by parts]
`= te^t - e^t = e^t (t - 1)`
`= e^(tan x) (tan x - 1)`
∴ From (1) `y.e^(tan x) = e^(tan x) (tan x - 1) + C`
⇒ ` y = (tan x - 1) + Ce^(-tan x),` Which is the required 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:
`x dy/dx + 2y= x^2 log x`
For the differential equation, find the general solution:
(1 + x2) dy + 2xy dx = cot x dx (x ≠ 0)
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.
`(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?
Solve the differential equation `x dy/dx + y = x cos x + sin x`, given that y = 1 when `x = pi/2`
x dy = (2y + 2x4 + x2) dx
\[\frac{dy}{dx}\] = y tan x − 2 sin x
\[\frac{dy}{dx}\] + 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 \[\left( y + 3 x^2 \right)\frac{dx}{dy} = x\]
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:-
\[\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)\]
Find the integerating factor of the differential equation `xdy/dx - 2y = 2x^2` .
Solve the differential equation: (1 +x2 ) dy + 2xy dx = cot x dx
Solve the following differential equation:
`("x" + 2"y"^3) "dy"/"dx" = "y"`
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 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.
Integrating factor of `dy/dx + y = x^2 + 5` is ______
Which of the following is a second order differential equation?
The integrating factor of differential equation `(1 - y)^2 (dx)/(dy) + yx = ay(-1 < y < 1)`
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 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 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
