Advertisements
Advertisements
प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Advertisements
उत्तर
y = c1e2x + c2e5x ....(1)
Differentiating twice w.r.t. x, we get
`"dy"/"dx" = "c"_1"e"^(2"x") xx 2 + "c"_2"e"^(5"x") xx 5`
∴ `"dy"/"dx" = 2"c"_1"e"^(2"x") + 5"c"_2"e"^(5"x")` ....(2)
and `("d"^2"y")/"dx"^2 = 2"c"_1"e"^(2"x") xx 2 + 5"c"_2"e"^(5"x") xx 5`
∴ `("d"^2"y")/"dx"^2 = 4"c"_1"e"^(2"x") + 25"c"_2"e"^("5x")` .....(3)
The equations (1), (2) and (3) are consistent in c1e2x and c2e5x
∴ determinant of their consistency is zero.
∴ `|("y",1,1),("dy"/"dx",2,5),(("d"^2"y")/"dx"^2,4,25)| = 0`
∴ y(50 - 20) - `1(25"dy"/"dx" - 5 ("d"^2"y")/"dx"^2) + 1 (4"dy"/"dx" - 2("d"^2"y")/"dx"^2) = 0`
∴ 30y - 25`"dy"/"dx" + 5("d"^2"y")/"dx"^2 + 4 "dy"/"dx" - 2("d"^2"y")/"dx"^2 = 0`
∴ `3("d"^2"y")/"dx"^2 - 21"dy"/"dx" + 30"y" = 0`
∴ `("d"^2"y")/"dx"^2 - 7"dy"/"dx" + 10"y" = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
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 - a)2 = 4(x - b)
Find the differential equation of the ellipse whose major axis is twice its minor axis.
Find the differential equation of all circles having radius 9 and centre at point (h, k).
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`
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
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
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 `("x + y")^2 "dy"/"dx" = 1` is
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 ______.
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")`
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 + "y"^2 = "r"^2; "x" "dy"/"dx" + "r" sqrt(1 + ("dy"/"dx")^2) = "y"`
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`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)`
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:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
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:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
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 solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Form the differential equation of family of standard circle
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
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 = ______.
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) ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
The differential equation for a2y = log x + b, 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.
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.
