Advertisements
Advertisements
प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
उत्तर
y = a sin (x + b)
∴ `"dy"/"dx" = "a" "d"/"dx" [sin ("x + b")]`
`= "a" cos ("x + b") - "d"/"dx" ("x + b")`
= a cos (x + b) × (1 + 0)
= a cos (x + b)
and `("d"^2 "y")/"dx"^2 = "a" "d"/"dx"[cos ("x + b")]`
`= "a" [- sin ("x + b")] * "d"/"dx"("x + b")`
`= - "a" sin ("x + b") xx (1 + 0)`
∴ `("d"^2 "y")/"dx"^2 = - "y"` .....[By (1)]
∴ `("d"^2 "y")/"dx"^2 + "y" = 0`
This is the required D.E.
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Form the differential equation of all parabolas whose axis is the X-axis.
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`
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = xm; `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
For the following differential equation find the particular solution satisfying the given condition:
`("x" + 1) "dy"/"dx" - 1 = 2"e"^-"y" , "y" = 0`, when x = 1
Reduce the following differential equation to the variable separable form and hence solve:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Choose the correct option from the given alternatives:
The differential equation of all circles having their centres on the line y = 5 and touching the X-axis is
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`
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Form the differential equation of y = (c1 + c2)ex
Find the differential equation of the family of all non-vertical lines in a plane
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be –8x, where A and B are arbitrary constants
The rate of disintegration of a radio active element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to ______.
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
Form the differential equation of all concentric circles having centre at the origin.
