Advertisements
Advertisements
प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)`
Advertisements
उत्तर
y = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)` ....(1)
Differentiating twice w.r.t. x, we get
`"dy"/"dx" = "Ae"^(3"x" + 1) * "d"/"dx" (3"x" + 1) + "Be"^(- 3"x" + 1) * "d"/"dx" (- 3"x" + 1)`
`= "Ae"^(3"x" + 1) xx (3 + 1 + 0) + "Be"^(- 3"x" + 1) xx (- 3 xx 1 + 0)`
`= 3"Ae"^(3"x" + 1) - 3 "Be"^(- 3"x" + 1)`
and `("d"^2 "y")/"dx"^2 = 3 "Ae"^(3"x" + 1) * "d"/"dx" ("3x" + 1) - 3 "Be"^(- 3"x" + 1) * "d"/"dx" (- 3"x" + 1)`
`= 3"Ae"^(3"x" + 1) xx (3 xx 1 + 0) - 3"Be"^(- 3"x" + 1) xx (- 3 xx 1 + 0)`
`= 9"Ae"^(3"x" + 1) - 9"Be"^(- 3"x" + 1)`
`= 9 ("Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1))`
= 9y ....[By (1)]
∴ `("d"^2"y")/"dx"^2 - 9"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:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = Ae5x + Be-5x
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 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:
`"dy"/"dx" = cos("x + y")`
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
In the following example verify that the given function is a solution of the differential equation.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
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
The general solution of `(dy)/(dx)` = e−x is ______.
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Form the differential equation of y = (c1 + c2)ex
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
The family of curves y = `e^("a" sin x)`, where a is an arbitrary constant, is represented by the differential equation.
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
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 general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Form the differential equation whose general solution is y = a cos 2x + b sin 2x.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
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
