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:
Ax2 + By2 = 1
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
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
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 = `(sin^-1 "x")^2 + "c"; (1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`
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:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
Solve the following differential equation:
`2"e"^("x + 2y") "dx" - 3"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:
(2x - 2y + 3)dx - (x - y + 1)dy = 0, when x = 0, y = 1.
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
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
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:
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`
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
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
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation from the relation x2 + 4y2 = 4b2
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 non-vertical lines in a plane
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
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 of the curve represented by xy = aex + be–x + x2
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation for a2y = log x + b, is ______.
Form the differential equation of all concentric circles having centre at the origin.
