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) = tan^2(x + y)`
Solve the following differential equation:
`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`
Solve the following differential equation:
`(1 + 3"e"^(y/x))"d"y + 3"e"^(y/x)(1 - y/x)"d"x` = 0, given that y = 0 when x = 1
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 solution of `("d"y)/("d"x) + "p"(x)y = 0` is
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) = y/x + (∅(y/x))/(∅(y/x))` is
Solve: `("d"y)/("d"x) + "e"^x + y"e"^x = 0`
Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) - y = sqrt(x^2 + y^2)`
Solve the following:
`x ("d"y)/("d"x) + 2y = x^4`
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:
Solution of `("d"x)/("d"y) + "P"x = 0`
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) + "P"y` = Q where P and Q are the function of x is
Choose the correct alternative:
The variable separable form of `("d"y)/("d"x) = (y(x - y))/(x(x + y))` by taking y = vx and `("d"y)/("d"x) = "v" + x "dv"/("d"x)` is
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Form the differential equation having for its general solution y = ax2 + bx
Solve x2ydx – (x3 + y3) dy = 0
