Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
उत्तर
The given differential equation can be written as
`("d"y)/("d"x) = (- 2xy)/(x^2 + 2y^2)` .........(1)
This is a homogenous differential equation
Putting y = vx
`("d"y)/("d"x) = "v"(1) + x "dv"/("d"x)`
`("d"y)/("d"x) = "v" + x "dv"/("d"x)`
(1) ⇒ `"v" + x "dv"/("d"x) = (-2x "v"x)/(x^2 + 2("v"x)^2)`
= `(- 2"v"x^2)/(x^2 + 2"v"^2x^2)`
= `(- x^2 2"v")/(x^2[1 + 2"v"^2])`
= `(- 2"v")/(1 + 2"v"^2)`
`"v" + x "dv"/("d"x) = (- 2"v")/(1 + 2"v"^2) - "v"`
`x "dv"/("d"x) = (- 2"v")/(1 + 2"v"^2) - "v"`
`x "dv"/("d"x) = (-2"v" - "v"(1 + 2"v"^2))/(1 + 2"v"^2)`
`x "dv"/("d"x) = (-2"v" - "v" - 2"v"^3)/(1 + 2"v"^2)`
`x "dv"/("d"x) = (-3"v" - "v"^3)/(1 + 2"v"^2)`
`int((1 + 2"v"^2)/(3"v" + 2"v"^3)) "dv" = - int ("d"x)/x`
Multiply and Divide by 3, we get
`1/3 int(3 + 6"v"^2)/(3"v" + "v"^3) "dv" = -int ("d"x)/x`
`1/3 log(3"v" + "v"^3) = - logx + log |"C"_1|`
`1/3 log(3"v" + 2"v"^3) + logx = log |"C"_1|`
log (3v + 2v3) + 3log (x) = 3 log (C1)
log (3v + 2v3) + log (x)3 = log (C1)3
log (3v + 2v3)x3 = log C13
(3v + 2v3)x3 = C13
`(3(y/x) + 2(y/x)^3)x^3` = C13
`((3y)/x + (2y^3)/x^3)x^3` = C13
`((3x^2y + 2y^3)x^3)/x^3` = C13
3x2y + 2y3 = C13
3x2y + 2y3 = C is a required solution.
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:
`(ydx - xdy) cot (x/y)` = ny2 dx
Solve the following differential equation:
`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0
Solve the following differential equation:
`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
Choose the correct alternative:
The solution of `("d"y)/("d"x) + "p"(x)y = 0` is
Solve: `("d"y)/("d"x) = "ae"^y`
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:
`x ("d"y)/("d"x) - y = sqrt(x^2 + y^2)`
Solve the following homogeneous differential equation:
An electric manufacturing company makes small household switches. The company estimates the marginal revenue function for these switches to be (x2 + y2) dy = xy dx where x represents the number of units (in thousands). What is the total revenue function?
Solve the following:
`("d"y)/("d"x) + y/x = x'e"^x`
Solve the following:
`("d"y)/("d"x) + y tan x = cos^3x`
Solve the following:
If `("d"y)/("d"x) + 2 y tan x = sin x` and if y = 0 when x = `pi/3` express y in term of x
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:
The integrating factor of the differential equation `("d"y)/("d"x) + "P"x` = Q is
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:
A homogeneous differential equation of the form `("d"y)/("d"x) = f(y/x)` can be solved by making substitution
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Solve x2ydx – (x3 + y3) dy = 0
