Advertisements
Advertisements
प्रश्न
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`
उत्तर
y = `(sin^-1 "x")^2 + "c"` .....(1)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx" (sin^-1 "x")^2 + 0`
∴ `"dy"/"dx" = 2(sin^-1 "x") * "d"/"dx" (sin^-1 "x")`
`= 2 sin^-1 "x" xx 1/sqrt(1 - "x"^2)`
∴ `sqrt(1 - "x"^2) "dy"/"dx" = 2 sin^-1 "x"`
∴ `(1 - "x"^2) ("dy"/"dx")^2 = 4(sin^-1 "x")^2`
∴ `(1 - "x"^2) ("dy"/"dx")^2 = 4("y - c")` ....[By (1)]
Differentiating again w.r.t. x, we get
`(1 - "x"^2) * "d"/"dx" ("dy"/"dx")^2 + ("dy"/"dx")^2 * "d"/"dx" (1 - "x"^2) = 4 "d"/"dx" ("y - c")`
∴ `(1 - "x"^2) * 2 "dy"/"dx" * ("d"^2"y")/"dx"^2 - 2"x" ("dy"/"dx")^2 = 4 ("dy"/"dx" - 0)`
Cancelling `2 "dy"/"dx"` throughout, we get
`(1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`
Hence, y = (sin-1 x)2 + c is a solution of the D.E.
`(1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`
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)2 = 4(x - b)
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.
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`
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`
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:
`2"e"^("x + 2y") "dx" - 3"dy" = 0`
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
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)
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 parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
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:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
Form the differential equation of y = (c1 + c2)ex
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
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
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
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 ______.
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
The differential equation for a2y = log x + b, is ______.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
