Advertisements
Advertisements
Question
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
Advertisements
Solution
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
RELATED QUESTIONS
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:
`(y^2 - 2xy) "d"x = (x^2 - 2xy) "d"y`
Solve the following differential equation:
`x ("d"y)/("d"x) = y - xcos^2(y/x)`
Choose the correct alternative:
The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` 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) = "ae"^y`
Solve: `log(("d"y)/("d"x))` = ax + by
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:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
Solve the following:
A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. A man invests ₹ 1,00,000 in the bank deposit which accrues interest, 8% per year compounded continuously. How much will he get after 10 years? (e0.8 = 2.2255)
Choose the correct alternative:
If y = ex + c – c3 then its differential equation is
Choose the correct alternative:
Solution of `("d"x)/("d"y) + "P"x = 0`
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:
A homogeneous differential equation of the form `("d"y)/("d"x) = f(y/x)` can be solved by making substitution
Choose the correct alternative:
A homogeneous differential equation of the form `("d"x)/("d"y) = f(x/y)` can be solved by making substitution
Form the differential equation having for its general solution y = ax2 + bx
Solve (x2 + y2) dx + 2xy dy = 0
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
