Advertisements
Advertisements
Question
Solve the following differential equation:
x cos y dy = ex(x log x + 1) dx
Advertisements
Solution
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
RELATED QUESTIONS
Solve the following differential equation:
(ey + 1)cos x dx + ey sin x dy = 0
Solve the following differential equation:
`("d"y)/("d"x) = tan^2(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
Choose the correct alternative:
The solution of `("d"y)/("d"x) = 2^(y - x)` is
Solve: `log(("d"y)/("d"x))` = ax + by
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) = x + y`
Solve the following homogeneous differential equation:
`(x - y) ("d"y)/("d"x) = x + 3y`
Solve the following:
`("d"y)/("d"x) - y/x = x`
Solve the following:
`("d"y)/(""dx) + y cos x = sin x cos x`
Solve the following:
`("d"y)/("d"x) + y/x = x'e"^x`
Solve the following:
`("d"y)/("d"x) + y/x = x"e"^x`
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x))` is
Form the differential equation having for its general solution y = ax2 + bx
Solve `("d"y)/("d"x) = xy + x + y + 1`
