Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
उत्तर
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
∴ `"dx"/"dy" + "2x" = 5/4"e"^(- 3"y")` .....(1)
This is the linear differential equation of the form
`"dx"/"dy" + "Px" = "Q"` where P = 2 and `"Q" = 5/4 "e"^(- 3"y")`
∴ I.F. = `"e"^(int "P dy") = "e"^(2 "dy") = "e"^("2y")`
∴ the solution of (1) is given by
`"x" * ("I.F.") = int "Q" * ("I.F.") "dy" + "c"_1`
∴ `"x" * "e"^(2"y") = int 5/4 "e"^(- 3"y") * "e"^"2y" "dy" + "c"_1`
∴ `"x" * "e"^(2"y") = 5/4 int "e"^-"y" "dy" + "c"_1`
∴ `"x" "e"^(2"y") = 5/4 * ("e"^-"y")/-1 + "c"_1`
∴ 4xe2y = - 5e-y + 4c1
∴ 4xe2y + - 5e-y = c, where c = 4c1
This is the general solution.
APPEARS IN
संबंधित प्रश्न
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 = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = c1e2x + c2e5x
Find the differential equation of all circles having radius 9 and centre at point (h, k).
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
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:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
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:
`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:
`(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 solution of `("x + y")^2 "dy"/"dx" = 1` 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"`
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` 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.
`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
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:
x dy = (x + y + 1) dx
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"`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
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
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 differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
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) ______.
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 ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Form the differential equation of all concentric circles having centre at the origin.
