Advertisements
Advertisements
प्रश्न
Solve the differential equation: (1 + x2) dy + 2xy dx = cot x dx
उत्तर
The given differential equation is
(1 + x2) dy + 2xy dx = cot x dx
`(d"y")/(d"x") + (2"xy")/(1 + "x"^2) = cot"x"/(1+"x"^2)`
This equation is a linear differential equation of the form:
`dy/dx + py = Q ( "where p" = (2x)/(1 + x^2) and Q = (cot x)/(1 + x^2) )`
`"IF" = e^(int pd"x") = e^(int(2"x")/(1+"x"^2) dx) = e^log(1 + "x"^2) = 1 + x^2`
The general solution of the given differential equation is given by the relation,
y( I.F.) = `int ( "Q" xx "I.F.") dx + C`
⇒ `y(1 + x^2) = int [ (cot x)/(1+ x^2) (1 + x^2)]dx + C`
⇒ `y(1 + x^2) = int cot x dx + c`
⇒ `y(1 + x^2) = log| sin x | + c`.
APPEARS IN
संबंधित प्रश्न
Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`
Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`
Solve the differential equation `sin^(-1) (dy/dx) = x + y`
Find the equation of the 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 decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
Solve the following differential equation :
`"dy"/"dx" + "y" = cos"x" - sin"x"`
Solve the differential equation `"dy"/"dx" + y/x` = x2.
`"dy"/"dx" + y` = 5 is a differential equation of the type `"dy"/"dx" + "P"y` = Q but it can be solved using variable separable method also.
`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.
Solution of the differential equation of the type `("d"x)/("d"y) + "p"_1x = "Q"_1` is given by x.I.F. = `("I"."F") xx "Q"_1"d"y`.
Correct substitution for the solution of the differential equation of the type `("d"x)/("d"y) = "g"(x, y)` where g(x, y) is a homogeneous function of the degree zero is x = vy.
Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.
The solution of the differential equation `"dy"/"dx" = "k"(50 - "y")` is given by ______.
Solve the differential equation:
`"dy"/"dx" = 2^(-"y")`
The solution of the differential equation `(dx)/(dy) + Px = Q` where P and Q are constants or functions of y, is given by
If α, β are different values of x satisfying the equation a cos x + b sinα x = c, where a, b and c are constants, then `tan ((alpha + beta)/2)` is
The solution of the differential equation `(dy)/(dx) = 1 + x + y + xy` when y = 0 at x = – 1 is
Let y = y(x) be the solution of the differential equation `(dy)/(dx) + (sqrt(2)y)/(2cos^4x - cos2x) = xe^(tan^-1(sqrt(2)cost2x)), 0 < x < π/2` with `y(π/4) = π^2/32`. If `y(π/3) = π^2/18e^(-tan^-1(α))`, then the value of 3α2 is equal to ______.
The population P = P(t) at time 't' of a certain species follows the differential equation `("dp")/("dt")` = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is ______.
The solution of the differential equation `(1 + y^2) + (x - e^(tan^-1y)) (dy)/(dx)` = 0, is ______.
