Advertisements
Advertisements
प्रश्न
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Advertisements
उत्तर
Equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis is
(x – 0)2 = 4a(y + 1)
x2 = 4a(y + 1) .......(1)
x2 = 4 ay + 4a
Differentiating equation (1) with respect to ‘x’, we get
2x = 4a y’
`(2x)/(y"'")` = 4a
Substituting 4a value in equation (1), we get
x² = `(2x)/(y"'") (y + 1)`
`x^2/x = 2/(y"'") (y + 1)`
x = `2/(y"'") (y + 1)`
xy’ = 2(y + 1)
xy’ = 2y + 2
xy’ – 2y – 2 = 0 is a required differential equation.
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = e-x + Ax + B; `"e"^"x" ("d"^2"y")/"dx"^2 = 1`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
In the following example verify that the given function is a solution of the differential equation.
`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
If m and n are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn - m + n = ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
