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:
y2 = (x + c)3
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
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:
c1x3 + c2y2 = 5
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:
y = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`
Solve the following differential equation:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
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:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
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 ______.
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"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
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 the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
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 solution of `("d"y)/("d"x)` = 1 is
Form the differential equation of family of standard circle
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 of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
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 differential equation of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
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 representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
