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`
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:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y2 = (x + c)3
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 the ellipse whose major axis is twice its minor axis.
Form the differential equation of all parabolas whose axis is the X-axis.
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`
Solve the following differential equation:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
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.
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
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 `("x + y")^2 "dy"/"dx" = 1` is
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
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.
`"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)
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
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:
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
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 general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
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 circles passing through the origin and having their centres on the x-axis
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
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 ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
