Advertisements
Advertisements
Question
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Advertisements
Solution
y log y = (log y2 - x) `"dy"/"dx"`
∴ `1/("dy"/"dx") = (log "y"^2 - "x")/("y" log "y")`
∴ `"dx"/"dy" = (2 log "y - x")/("y" log "y")`
∴ `"dx"/"dy" = 2/"y" - "x"/("y" log "y")`
∴ `"dx"/"dy" + "x"/("y" log "y") = 2/"y"` .....(1)
This is the linear differential equation of the form
`"dx"/"dy" + "Px" = "Q"` where P = `1/("y" log "y") and "Q" = 2/"y"`
∴ I.F. = `"e"^(int "P dy") = "e"^(int 1/("y" log "y")"dy")`
`= "e"^(int (1//"y")/(log "y")"dy") = "e"^(log |log "y"|)` = log y
∴ the solution of (1) is given by
`"x" * ("I.F.") = int "Q" * ("I.F.") "dy" + "c"`
∴ `"x" * log "y" = int 2/"y" * log "y" "dy" + "c"`
∴ `(log "y") * "x" = 2 int (log "y")/"y" "dy" + "c"`
Put log y = t
∴ `1/"y" "dy" = "dt"`
∴ `("log y")*"x" = 2 int "t" "dt" + "c"`
∴ x log y = `2 * "t"^2/2 + "c"`
∴ x log y = (log y)2 + c
This is the general solution.
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:
Ax2 + By2 = 1
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)
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
Solve the following differential equation:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"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.
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`
Reduce the following differential equation to the variable separable form and hence solve:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
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 ______.
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`
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"))`
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" = ("2y" - "x")/("2y + x")`
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
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
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Form the differential equation of y = (c1 + c2)ex
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
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
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 corresponding to the family of curves represented by the equation y = Ae8x + Be –8x, where A and B are arbitrary constants
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
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 ______.
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 the family of circles touching Y-axis at the origin is ______.
The differential equation for a2y = log x + b, is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
