Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
The given differential equation may be written as
`("d"y)/("d"x) = (-3"e"^(y/x)(1 - y/x))/((1 + 3"e"^(y/x))` ......(1)
This is a homogeneous differential equation,
Putting y = vx
⇒ `("d"y)/("d"x) = "v"(1) + x "dv"/("d"x)`
(1) ⇒ `"v" + x "dv"/("d"x) = (-3"e"^(y/x)(1 - y/x))/(1 + 3"e"^(y / x))`
= `(- 3"e"^((vx)/x) (1 - "v"))/(1 + 3"e"^((vx)/x)`
= `(- 3"e"^"v"(1 - "v"))/(1 + 3"e"^((vx)/x)`
`x "dv"/("d"x) = (- 3"e"^"v" + 3"e"^"v" "v")/((1 + 3"e"^"v")) - "v"`
= `(- 3"e"^"v" + 3"e"^"v" "v" - "v"(1 + 3"e"^"v"))/(1 + 3"e"^"v")`
= `(- 3"e"^"v" + 3"e"^"v" "v" - "v" - 3"e"^"v" "v")/(1 + 3"e"^"v")`
`x "dv"/("d"x) = (- 3"e"^"v" - "v")/(1 + 3"e"^"v")`
`((1 + 3"e"^"v"))/(- 3"e"^"v" - "v") "dv" = ("d"x)/x`
`- ((1 + 3"e"^"v"))/(("v" + 3"e"^"v")) "dv" = ("d"x)/x`
`- int ((1 + 3"e"^"v"))/("v" + 3"e"^"v") "dv" - int ("d"x)/x = log ("c")`
`- int ((1 + 3"e"^"v"))/("v" + 3"e"^"v") "dv" + int ("d"x)/x = log ("c")`
`log("v" + 3"e"^"v") + log(x) = log("c")`
`log ("v" + 3"e"^"v")x = log "c"`
`x("v" + 3"e"^"v") = "c"`
`x(y/x + 3"e"^(y/x))` = c .........`(∵ "v" = y/x)`
`(xy)/x + 3x"e"^(y/x)` = c
`y + 3x"e"^(y/x)` = c
Given that y = 0 when x = 1
0 + 3(1) e° = c
3 = c
∴ `y + 3x"e"^(y/x)` = 3 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:
`tan y ("d"y)/("d"x) = cos(x + y) + cos(x - y)`
Solve the following differential equation:
`("d"y)/("d"x) = tan^2(x + y)`
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
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
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`
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:
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:
Which of the following is the homogeneous differential equation?
Form the differential equation having for its general solution y = ax2 + bx
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
