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:
`y"d"x + (1 + x^2)tan^-1x "d"y`= 0
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 + y cos(y/x)] "d"x = x cos(y/x) "d"y`
Solve the following differential equation:
`(x^3 + y^3)"d"y - x^2 y"d"x` = 0
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
Choose the correct alternative:
The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` 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: `("d"y)/("d"x) + "e"^x + y"e"^x = 0`
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:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
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 differential equation of y = mx + c is (m and c are arbitrary constants)
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:
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 `("d"y)/("d"x) + y cos x + x = 2 cos x`
