Advertisements
Advertisements
Question
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Advertisements
Solution
y = (c1 + c2x)ex ......(i)
Here, c1 and c2 are arbitrary constants.
Differentiating w.r.t. x, we get
`("d"y)/("d"x)` = (c1 + c2x)ex + c2ex
∴ `("d"y)/("d"x)` = y + c2ex ......(ii) .......[From(i)]
Again, differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = ("d"y)/("d"x) + "c"_2"e"^x`
∴ c2ex = `("d"^2y)/("d"x^2) - ("d"y)/("d"x)` .....(iii)
Substituting (iii) in (ii), we get
`("d"y)/("d"x) = y + ("d"^2y)/("d"x^2) - ("d"y)/("d"x)`
∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0
APPEARS IN
RELATED QUESTIONS
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:
y2 = (x + c)3
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Find the differential equation of the ellipse whose major axis is twice its minor 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 = 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:
`"y" - "x" "dy"/"dx" = 0`
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:
`2"e"^("x + 2y") "dx" - 3"dy" = 0`
For the following differential equation find the particular solution satisfying the given condition:
3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.
For the following differential equation find the particular solution satisfying the given condition:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 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
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
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
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
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 solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Form the differential equation of y = (c1 + c2)ex
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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 circles passing through the origin and having their centres on the x-axis
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
Find the differential equation of the curve represented by xy = aex + be–x + x2
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 ______.
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
The differential equation for a2y = log x + b, is ______.
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
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.
