Advertisements
Advertisements
प्रश्न
In the following example verify that the given function is a solution of the differential equation.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Advertisements
उत्तर
`"y" = 3 "cos" (log "x") + 4 sin (log "x")` ...(1)
Differentiating both sides w.r.t. x, we get
`"dy"/"dx" = 3 "d"/"dx" [cos (log "x")] + 4 "d"/"dx" [sin (log "x")]`
`= 3 [- sin (log "x")] "d"/"dx" (log "x") + 4 cos (log "x") "d"/"dx" (log "x")`
`= - 3 sin (log "x") xx 1/"x" + 4 cos (log "x") xx 1/"x"`
∴ `"x" "dy"/"dx" = - 3 sin (log "x") + 4 cos (log "x")`
Differentiating again w.r.t. x, we get,
`"x" "d"/"dx" ("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("x") = - 3 "d"/"dx" [sin (log "x")] + 4 "d"/"dx"[cos (log "x")]`
∴ `"x" ("d"^2"y")/"dx"^2 + "dy"/"dx" xx 1 = - 3 cos (log "x") * "d"/"dx" (log "x") + 4 [- sin (log "x")]* "d"/"dx" (log "x")`
∴ `"x" ("d"^2"y")/"dx"^2 + "dy"/"dx" = - 3 cos (log "x") xx 1/"x" - 4 sin (log "x") xx 1/"x"`
∴ `"x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" = - [3 cos (log "x") + 4 sin (log "x")] = - "y"` ...[By (1)]
∴ `"x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Hence, y = 3 cos (log x) + 4 sin (log x) is a solution of the D.E. `"x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
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:
y = A cos (log x) + B sin (log x)
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
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 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:
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 = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "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"`
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
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`
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)
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
The general solution of `(dy)/(dx)` = e−x is ______.
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 solution of `("d"y)/("d"x)` = 1 is
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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 equations of the family of all the ellipses having foci on the y-axis and centre at the origin
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
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 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 ______.
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.
