Advertisements
Advertisements
Question
Solve the following differential equation:
`y"d"x + (1 + x^2)tan^-1x "d"y`= 0
Advertisements
Solution
`y"d"x + (1 + x^2)tan^-1x "d"y`
Take t = tan–1x
dt `1/(1 + x^2) "d"x`
The equation can be written as
`("d"x)/((1 + x^2)tan^-) = - ("d"y)/y`
`"dt"/"t" = - ("d"y)/y`
Taking Integration on both sides, we get
`int "dt"/"t" = int ("d"y)/y`
log t = – log y + log C
log(tan–1x) = – log y + log C
log (y(tan–1x)) + log y = log C
y tan–1x = C
APPEARS IN
RELATED QUESTIONS
If F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by `"M""dv"/"dt"` = F – kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0
The velocity v, of a parachute falling vertically satisfies the equation `"v" (dv)/(dx) = "g"(1 - v^2/k^2)` where g and k are constants. If v and are both initially zero, find v in terms of x
Solve the following differential equation:
`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`
Solve the following differential equation:
(ey + 1)cos x dx + ey sin x dy = 0
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
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 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
Solve: `(1 + x^2)/(1 + y) = xy ("d"y)/("d"x)`
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:
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
Choose the correct alternative:
If y = ex + c – c3 then its differential equation is
Choose the correct alternative:
The differential equation of x2 + y2 = a2
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 (D2 – 3D + 2)y = e4x given y = 0 when x = 0 and x = 1
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
Solve x2ydx – (x3 + y3) dy = 0
Solve `("d"y)/("d"x) = xy + x + y + 1`
