Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
x cos y dy = ex(x log x + 1) dx
Advertisements
उत्तर
The equation can be written as
cos y dy = `"e"^x ((xlogx + 1))/x "d"x`
cos y dy = `"e"^x [(xlogx)/x + 1/x] "d"x`
cos y dy = `"e"^x [logx + 1/x] "d"x`
Taking integration on both sides, we get
`int cos y "d"y = int "e"^x [log x + 1/x] "d"x` ........(1)
R.H.S
`int "e"^x [log x + 1/x] "d"x`
⇒ Take f(x) = log x
f'(x) = `1/x`
This of the form `int "e"^x ["f"(x) + "f'"(x)] "d"x = "e"^x "f"(x) + "C"`
∴ `int "e"^x [log x + 1/x] "d"x = "e"^x lo x + "C"`
Substituting in (1), we get
sin y = ey log x + C
APPEARS IN
संबंधित प्रश्न
Solve the following differential equation:
(ey + 1)cos x dx + ey sin x dy = 0
Solve the following differential equation:
`(ydx - xdy) cot (x/y)` = ny2 dx
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
Choose the correct alternative:
The solution of `("d"y)/("d"x) = 2^(y - x)` 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:
If sin x is the integrating factor of the linear differential equation `("d"y)/("d"x) + "P"y = "Q"`, then P is
Solve: `("d"y)/("d"x) = "ae"^y`
Solve: `y(1 - x) - x ("d"y)/("d"x)` = 0
Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0
Solve the following homogeneous differential equation:
`(x - y) ("d"y)/("d"x) = x + 3y`
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:
`("d"y)/(""dx) + y cos x = sin x cos x`
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:
The solution of the differential equation `("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x))` is
Solve (x2 + y2) dx + 2xy dy = 0
Solve `x ("d"y)/(d"x) + 2y = x^4`
A manufacturing company has found that the cost C of operating and maintaining the equipment is related to the length ’m’ of intervals between overhauls by the equation `"m"^2 "dC"/"dm" + 2"mC"` = 2 and c = 4 and when = 2. Find the relationship between C and m
Solve `("d"y)/("d"x) = xy + x + y + 1`
