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
Solve the following differential equation:
`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`
Solve the following differential equation:
`tan y ("d"y)/("d"x) = cos(x + y) + cos(x - y)`
Solve the following differential equation:
`(x^3 + y^3)"d"y - x^2 y"d"x` = 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 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
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: `("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:
`x ("d"y)/("d"x) - y = sqrt(x^2 + y^2)`
Solve the following homogeneous differential equation:
The slope of the tangent to a curve at any point (x, y) on it is given by (y3 – 2yx2) dx + (2xy2 – x3) dy = 0 and the curve passes through (1, 2). Find the equation of the curve
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:
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 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:
A homogeneous differential equation of the form `("d"x)/("d"y) = f(x/y)` can be solved by making substitution
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
Solve (D2 – 3D + 2)y = e4x given y = 0 when x = 0 and x = 1
