Advertisements
Advertisements
Question
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Advertisements
Solution
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"` ....(1)
This is the linear differential equation of the form
`"dy"/"dx" + "Py" = "Q",` where P = cot x and Q = x2 cot x + 2x
∴ I.F. = `"e"^(int "P dx") = "e"^(int "cot x" "dx")`
`= "e"^(log ("sin x")) = sin "x"`.
∴ the solution of (1) is given by
`"y" * ("I.F.") = int "Q" * ("I.F.")"dx" + "c"`
∴ y sin x = `int ("x"^2 "cot x" + "2x") "sin x dx" + "c"`
∴ y sin x = `int ("x"^2 cot "x" * sin "x" + "2x" sin "x") "dx" + "c"`
∴ y sin x = `int "x"^2 cos "x" "dx" + 2 int "x" sin "x" "dx" + "c"`
∴ y sin x = `"x"^2 int "cos x dx" - int["d"/"dx" ("x"^2) int "cos x dx"] "dx" + 2 int "x sin x dx" + "c"`
`= "x"^2 (sin "x") - int "2x" (sin "x")"dx" + 2 int "x sin x dx" + "c"`
`= "x"^2 sin x - 2 int "x" sin "x" "dx" + 2 int "x sin x dx" + "c"`
∴ y sin x = x2 sin x + c
∴ y = x2 + c cosec x
This is the general solution.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
Form the differential equation of all parabolas whose axis is the X-axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
Solve the following differential equation:
`"y" - "x" "dy"/"dx" = 0`
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
For the following differential equation find the particular solution satisfying the given condition:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
Choose the correct option from the given alternatives:
The differential equation of y = `"c"^2 + "c"/"x"` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 + "y"^2 = "r"^2; "x" "dy"/"dx" + "r" sqrt(1 + ("dy"/"dx")^2) = "y"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
The family of curves y = `e^("a" sin x)`, where a is an arbitrary constant, is represented by the differential equation.
Find the differential equation of the family of all non-vertical lines in a plane
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
Find the differential equation of the curve represented by xy = aex + be–x + x2
The differential equation of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Form the differential equation of all concentric circles having centre at the origin.
