Advertisements
Advertisements
Question
Solve the differential equation `"dy"/"dx" + y/x` = x2.
Solution
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
RELATED QUESTIONS
Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
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 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.
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.
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
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.
`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.
Correct substitution for the solution of the differential equation of the type `("d"y)/("d"x) = "f"(x, y)`, where f(x, y) is a homogeneous function of zero degree is y = vx.
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 ______.
The solution of the differential equation `(dy)/(dx) = 1 + x + y + xy` when y = 0 at x = – 1 is
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.
Solve the differential equation: xdy – ydx = `sqrt(x^2 + y^2)dx`
Find the general solution of the differential equation: (x3 + y3)dy = x2ydx
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 ______.
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 `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1. Then, the value of (y(3))2 is equal to ______.
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 ______.
