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:
x3 + y3 = 4ax
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
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`
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:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
Solve the following differential equation:
`"y" - "x" "dy"/"dx" = 0`
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:
`("x - y")^2 "dy"/"dx" = "a"^2`
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^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 solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
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.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
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 differential equation of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
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.
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
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 ______.
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.
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
Form the differential equation of all concentric circles having centre at the origin.
