Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
Advertisements
उत्तर
The given equation can be written as
`("d"x)/("d"y) = (x"e"^(x/y) + y)/(y"e"^(x/y))` ........(1)
This is a Homogeneous differential equation
Put x = vy
⇒ `("d"x)/("d"y) = "v" + y * "dv"/("d"y)`
(1) ⇒ `"v" + y * "dv"/("d"y) = ("vve"^"v" + y)/(y"e"^"v")`
`"v" + y * "dv"/("d"y) = (y("ve"^"v" + 1))/(y"e"^"v")`
`y "d"/("d"y) = ("ve"^"v" + 1)/"e"^"v" - "v"`
`y "dv"/("d"y) = ("ve"^"v" + 1 - "ve"^"v")/"e"^"v"`
`y "dv"/("d"y) = 1/"e"^"v"`
Seperating the variables
`int "e"^"v" "dv" = int ("d"y)/y`
ev = log y + log c
ev = log |cy|
i.e., `"e"^(x/y)` = log |cy| .......`[∵ "v" = x/y]`
APPEARS IN
संबंधित प्रश्न
The velocity v, of a parachute falling vertically satisfies the equation `"v" (dv)/(dx) = "g"(1 - v^2/k^2)` where g and k are constants. If v and are both initially zero, find v in terms of x
Solve the following differential equation:
`("d"y)/("d"x) = sqrt((1 - y^2)/(1 - x^2)`
Solve the following differential equation:
(ey + 1)cos x dx + ey sin x dy = 0
Solve the following differential equation:
`(ydx - xdy) cot (x/y)` = ny2 dx
Solve the following differential equation:
`tan y ("d"y)/("d"x) = cos(x + y) + cos(x - y)`
Solve the following differential equation:
(x2 + y2) dy = xy dx. It is given that y (1) = y(x0) = e. Find the value of x0
Choose the correct alternative:
The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` is
Choose the correct alternative:
The number of arbitrary constants in the particular solution of a differential equation of third order is
Solve: `(1 + x^2)/(1 + y) = xy ("d"y)/("d"x)`
Solve: ydx – xdy = 0 dy
Solve: `("d"y)/("d"x) + "e"^x + y"e"^x = 0`
Find the curve whose gradient at any point P(x, y) on it is `(x - "a")/(y - "b")` and which passes through the origin
Solve the following homogeneous differential equation:
(y2 – 2xy) dx = (x2 – 2xy) dy
Solve the following:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
Solve the following:
`("d"y)/("d"x) + y/x = x'e"^x`
Choose the correct alternative:
If sec2 x is an integrating factor of the differential equation `("d"y)/("d"x) + "P"y` = Q then P =
Choose the correct alternative:
The differential equation of x2 + y2 = a2
Choose the correct alternative:
A homogeneous differential equation of the form `("d"y)/("d"x) = f(y/x)` can be solved by making substitution
Solve (x2 + y2) dx + 2xy dy = 0
