Advertisements
Advertisements
Question
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.
Solution
According to the question,
\[\frac{dy}{dx} = \frac{1}{2y}\]
\[ \Rightarrow 2y dy = dx\]
Integrating both sides, we get
\[2\int y dy = \int dx\]
\[ \Rightarrow y^2 = x + C\]
Since the curve passes throught the point (4, 3), it satisfies the equation of the curve.
\[9 = 4 + C\]
\[ \Rightarrow C = 5\]
Putting the value of C in the equation of the curve, we get
\[ y^2 = x + 5\]
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`
Solve the differential equation `sin^(-1) (dy/dx) = x + y`
\[\frac{dy}{dx}\] + y tan x = cos x
\[\frac{dy}{dx}\] + y cot x = x2 cot x + 2x
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.
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.
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"`
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.
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
`int cos(log x) dx = F(x) + C` where C is arbitrary constant. Here F(x) =
If `x (dy)/(dx) = y(log y - log x + 1)`, then the solution of the dx equation 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 ______.
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, `(x^2 + 1)^2 ("dy")/("d"x) + 2x(x^2 + 1)"y"` = 1, such that y(0) = 0. If `sqrt("ay")(1) = π/32` then the value of 'a' is ______.
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
