Advertisements
Advertisements
प्रश्न
For the differential equation, find the general solution:
`x log x dy/dx + y= 2/x log x`
उत्तर
The given equation
`x log x dy/dx + y = 2/x log x`
or `dy/dx + y/(x log x) = 1/x^2` ....(i)
Comparing with `dy/dx + Py = Q`,
`P = 1/(x log x)` and `Q = 2/x^2`
∴ `I.F. = e^(intP dx) = e^(int 1/(x log x)dx)`
`= e^(log(log x)) = log x`
Hence the required solution
∴ `y × I.F. = int I.F. xx Q dx + C`
`=> y log x = 2 int 1/x^2 (log x) dx + C`
`=> y log x = 2 [log x (- 1/x) - int 1/x ((- 1)/x) dx] + C`
`=> y log x = (- 2)/x log x + 2 int 1/x^2 dx + C`
`=> y log x = (- 2)/x log x - 2/x + C`
⇒ `y = log x = (- 2)/x (1 + log |x|) + C`
APPEARS IN
संबंधित प्रश्न
For the differential equation, find the general solution:
`dy/dx + 3y = e^(-2x)`
For the differential equation, find the general solution:
`(x + 3y^2) dy/dx = y(y > 0)`
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:
`(1 + x^2)dy/dx + 2xy = 1/(1 + x^2); y = 0` when x = 1
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 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 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
(x + tan y) dy = sin 2y 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 \[\frac{dy}{dx} - y = \cos x\]
Find the integerating factor of the differential equation `x(dy)/(dx) - 2y = 2x^2`
If f(x) = x + 1, find `"d"/"dx"("fof") ("x")`
Solve the following differential equation:
`"x" "dy"/"dx" + "2y" = "x"^2 * log "x"`
Solve the following differential equation:
`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`
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"`.
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 integrating factor of the differential equation (1 + x2)dt = (tan-1 x - t)dx is ______.
Which of the following is a second order differential equation?
State whether the following statement is true or false.
The integrating factor of the differential equation `(dy)/(dx) + y/x` = x3 is – x.
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 ______.
The solution of the differential equation `dx/dt = (xlogx)/t` is ______.
If sec x + tan x is the integrating factor of `dy/dx + Py` = Q, then value of P 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 ______.
