Advertisements
Advertisements
Question
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`
Solution
y = xm
Differentiating twice w.r.t. x, we get
`"dy"/"dx" = "d"/"dx" ("x"^"m") = "mx"^("m - 1")`
and `("d"^2"y")/"dx"^2 = "d"/"dx" ("mx"^("m - 1")) = "m" "d"/"dx" ("x"^("m - 1")) = "m"("m" - 1) "x"^("m - 2")`
∴ `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my"`
`= "x"^2 * "m"("m" - 1) "x"^("m - 2") - "mx" * "mx"^("m" - 1) + "m" * "x"^"m"`
`= "m"("m - 1") "x"^"m" - "m"^2 "x"^"m" + "mx"^"m"`
`= ("m"^2 - "m" - "m"^2 + "m")"x"^"m" = 0`
This shows that y = xm is a solution of the D.E.
`"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0`.
APPEARS IN
RELATED QUESTIONS
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:
y2 = (x + c)3
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
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.
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
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
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 ______.
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 = "2y"^2 log "y", "x"^2 + "y"^2 = "xy" "dx"/"dy"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
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:
`"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:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
Find the differential equation of family of lines making equal intercepts on coordinate axes
Form the differential equation of family of standard circle
Find the differential equation from the relation x2 + 4y2 = 4b2
The differential equation having y = (cos-1 x)2 + P (sin-1 x) + Q as its general solution, where P and Q are arbitrary constants, is
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 parabolas with vertex at (0, –1) and having axis along the y-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
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 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 y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation of all parabolas whose axis is 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.
