Advertisements
Advertisements
Question
Solve the following:
`("d"y)/("d"x) - y/x = x`
Advertisements
Solution
Given `("d"y)/("d"x) + ((-1)/x)y = x`
It is of the form `("d"y)/("d"x) + "p"y` = Q
Here P = `(-1)/x, "Q"` = x
`int "p" "d"x = int (-1)/x "d"x`
= `- log x`
= `log(1/x)`
I.F = `"e"^(int pdx)`
= `"e"^(log(1/x)`
= `1/x`
The required solution is
y(I.F) = `int "Q"("I.F") "d"x + "c"`
`y(1/x) = int x(1/x) "d"x + "c"`
`y/x = int "d"x + "c"`
∴ `y/x` = x + c
APPEARS IN
RELATED QUESTIONS
Solve the following differential equation:
`sin ("d"y)/("d"x)` = a, y(0) = 1
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
Solve the following differential equation:
(x2 + y2) dy = xy dx. It is given that y (1) = y(x0) = e. Find the value of x0
Solve: `("d"y)/("d"x) = "ae"^y`
Solve: (1 – x) dy – (1 + y) dx = 0
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) = x + y`
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:
`x ("d"y)/("d"x) + 2y = x^4`
Form the differential equation having for its general solution y = ax2 + bx
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
