Advertisements
Advertisements
प्रश्न
Solve the differential equation `"dy"/"dx" + y/x` = x2.
Advertisements
उत्तर
The equation is of the type `"dy"/"dx" + "Py"` = Q, which is a linear differential equation.
Now I.F. = `int 1/x "d"x`
= elogx = x.
Therefore, solution of the given differential equation is
y.x = `int x x^2 "d"x`
i.e. yx = `x^4/4 + "c"`
Hence y = `x^3/4 + "c"/x`.
APPEARS IN
संबंधित प्रश्न
Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`
Solve `sin x dy/dx - y = sin x.tan x/2`
\[\frac{dy}{dx}\] + y tan x = cos x
The slope of the tangent to the curve at any point is the reciprocal of twice the ordinate at that point. The curve passes through the point (4, 3). Determine its equation.
A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet hung in the wind loses half of its moisture during the first hour, when will it have lost 95% moisture, weather conditions remaining the same.
Solve the differential equation : `"x"(d"y")/(d"x") + "y" - "x" + "xy"cot"x" = 0; "x" != 0.`
Solve the following differential equation :
`"dy"/"dx" + "y" = cos"x" - sin"x"`
`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.
Integrating factor of the differential equation of the form `("d"x)/("d"y) + "P"_1x = "Q"_1` is given by `"e"^(int P_1dy)`.
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.
If ex + ey = ex+y, then `"dy"/"dx"` is:
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 ______.
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.
Which of the following solutions may be used to find the number of children who have been given the polio drops?
Solve the differential equation:
`"dy"/"dx" = 2^(-"y")`
If `x (dy)/(dx) = y(log y - log x + 1)`, then the solution of the dx equation is
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Find the general solution of the differential equation: (x3 + y3)dy = x2ydx
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 ______.
If y = y(x) is the solution of the differential equation `(1 + e^(2x))(dy)/(dx) + 2(1 + y^2)e^x` = 0 and y(0) = 0, then `6(y^'(0) + (y(log_esqrt(3))))^2` 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 ______.
Let y = y(x) be the solution of the differential equation `xtan(y/x)dy = (ytan(y/x) - x)dx, -1 ≤ x ≤ 1, y(1/2) = π/6`. Then the area of the region bounded by the curves x = 0, x = `1/sqrt(2)` and y = y(x) in the upper half plane is ______.
Let y = y(x) be the solution of the differential equation, `(2 + sinxdy)/(y + 1) (dy)/(dx)` = –cosx. If y > 0, y(0) = 1. If y(π) = a, and `(dy)/(dx)` at x = π is b, then the ordered pair (a, b) is equal to ______.
If y = f(x), f'(0) = f(0) = 1 and if y = f(x) satisfies `(d^2y)/(dx^2) + (dy)/(dx)` = x, then the value of [f(1)] is ______ (where [.] denotes greatest integer function)
The solution of the differential equation `(1 + y^2) + (x - e^(tan^-1y)) (dy)/(dx)` = 0, is ______.
Solve the differential equation:
`dy/dx` = cosec y
