Advertisements
Advertisements
Question
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`
Advertisements
Solution
`"y" = "e"^"ax" sin "bx"`
∴ `"dy"/"dx" = "d"/"dx"("e"^"ax" sin "bx")`
`= "e"^"ax" * "d"/"dx" (sin "bx") + sin "bx" * "d"/"dx" ("e"^"ax")`
`= "e"^"ax" xx cos "bx" * "d"/"dx" ("bx") + sin "bx" * "e"^"ax" * "d"/"dx" ("ax")`
`= "e"^"ax" cos "bx" xx "b" + "e"^"ax" sin "bx" xx "a"`
`= "e"^"ax" ("b" cos "bx" + "a" sin "bx")`
and `("d"^2"y")/"dx"^2 = "d"/"dx"["e"^"ax" ("b" cos "bx" + "a" sin "bx")]`
`= "e"^"ax" * "d"/"dx" ("b" cos "bx" + "a" sin "bx") + ("b" cos "bx" + "a" sin "bx")*"d"/"dx"("e"^"ax")`
`= "e"^"ax" ["b" (- sin "bx") * "d"/"dx" ("bx") + "a" cos "bx" * "d"/"dx"("bx")] + ("b" cos "bx" + "a" sin "bx") * "e"^"ax" * "d"/"dx" ("ax")`
`= "e"^"ax" [- "b" sin "bx" xx "b" + "a" cos "bx" xx "b"] + ("b" cos "bx" + "a" sin bx) * "e"^"ax" xx "a"`
`= "e"^"ax" (- "b"^2 sin "bx" + "ab" cos "bx" +"ab" cos "bx" + "a"^2 sin "bx")`
`= "e"^"ax" [("a"^2 - "b"^2) sin "bx" + 2"ab" cos "bx"]`
∴ `("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y"`
`= "e"^"ax" [("a"^2 - "b"^2)] sin "bx" + 2"ab" cos "bx" - 2"a" * "e"^"ax" ("b" cos "bx" + "a" sin "bx") + ("a"^2 + "b"^2) * "e"^"ax" sin "bx"`
`= "e"^"ax" [("a"^2 - "b"^2) sin "bx" + 2"ab" cos "bx" - 2"ab" cos "bx" - 2"a"^2 sin "bx" + ("a"^2 + "b"^2) sin "bx"]`
`= "e"^"ax" [("a"^2 - "b"^2) sin "bx" - ("a"^2 - "b"^2) sin "bx"]`
`= "e"^"ax" xx 0 = 0`
Hence. y = `"e"^"ax"` sin bx is a solution of the D.E.
`("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
APPEARS IN
RELATED QUESTIONS
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 = Ae5x + Be-5x
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
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"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Solve the following differential equation:
`"y" - "x" "dy"/"dx" = 0`
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 0
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
Choose the correct option from the given alternatives:
The differential equation of y = `"c"^2 + "c"/"x"` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
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.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
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)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"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:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
The general solution of `(dy)/(dx)` = e−x is ______.
Form the differential equation of family of standard circle
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
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
Find the differential equation of the family of all non-horizontal lines in a plane
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 equation of the family of parabolas with vertex at (0, –1) and having axis along the y-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 ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
