Advertisements
Advertisements
प्रश्न
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Advertisements
उत्तर
`("x + 2y"^2) "dy"/"dx" = "y"`
∴ `"dy"/"dx" = ("x + 2y"^2)/"y" = "x"/"y" + "2y"`
∴ `"dx"/"dy" - 1/"y" * "x" = "2y"` ....(1)
This is the linear differential equation of the form
`"dx"/"dy" + "Px" = "Q"` where P = `- 1/"y"` and Q = 2y.
∴ I.F. = `"e"^(int "P dy") = "e"^(int - 1/"y" "dy")`
`= "e"^(- log "y") = "e"^(log (1/"y")) = 1/"y"`
∴ the solution of (1) is given by
`"x" * ("I.F.") = int "Q" * ("I.F.") "dy" + "c"`
∴ `"x" xx 1/"y" = int "2y" xx 1/"y" "dy" + "c"`
∴ `"x"/"y" = 2 int 1 "dy" + "c"`
∴ `"x"/"y" = 2"y" + "c"`
∴ x = 2y2 + cy
This is the general solution.
When x = 2, y = 1, we have
2 = 2(1)2 + c(1)
∴ c = 0
∴the particular solution is x = 2y2.
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y2 = (x + c)3
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
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"`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
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:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 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 `("x + y")^2 "dy"/"dx" = 1` is
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
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 = `"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:
`"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")`
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
Find the differential equation from the relation x2 + 4y2 = 4b2
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 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 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 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 ______.
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)` = ______.
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 ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
