Advertisements
Advertisements
Question
For the following differential equation find the particular solution satisfying the given condition:
3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.
Advertisements
Solution
3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.
∴`(3"e"^"x")/(1 + "e"^x) "dx" + ("sec"^2"y")/("tan y") "dy" = 0`
Integrating both sides, we get
`3 int "e"^"x"/(1 + "e"^"x") "dx" + int (sec^2"y")/(tan "y") "dy" = "c"_1`
Each of these integrals is of the type
`int ("f"'("x"))/("f"("x")) "dx" = log |"f"(x)| + "c"`
∴ the general solution is
3 log |1 + ex| + log |tan y| = log c, where c1 =log c
∴ log |(1 + ex)3 * tan y| = log c
∴ (1 + ex)3 tan y = c
When x = 0, y = π, we have
(1 + e0)3 tan π = c
∴ c = 0
∴ the particular solution is (1 + ex)3 tan y = 0
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = e-x + Ax + B; `"e"^"x" ("d"^2"y")/"dx"^2 = 1`
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
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:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Reduce the following differential equation to the variable separable form and hence solve:
(2x - 2y + 3)dx - (x - y + 1)dy = 0, when x = 0, y = 1.
Choose the correct option from the given alternatives:
The solution of `("x + y")^2 "dy"/"dx" = 1` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
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"`
In the following example verify that the given function is a solution of the differential equation.
`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
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:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
The general solution of `(dy)/(dx)` = e−x is ______.
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Form the differential equation of family of standard circle
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Find the differential equation from the relation x2 + 4y2 = 4b2
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 equations of the family of all the ellipses having foci on the y-axis and centre at the origin
The rate of disintegration of a radio active element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to ______.
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
If m and n are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn - m + n = ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
Form the differential equation whose general solution is y = a cos 2x + b sin 2x.
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
