Advertisements
Advertisements
Question
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
Advertisements
Solution
Given the equation of family of parabolas with latus rectum 4a and axes are parallel to x-axis then
(y – b)2 = 4a(x – a), where (a, b) is the vertex of parabola.
y2 – 2yb + b2 = 4ax – 4a2 ........(1)
Differentiating equation (1) with respect to x, we get
`2y ("d"y)/("d"x) - 2"b" ("d"y)/("d"x) + 0 = 4"a" - 0`
`2(y ("d"y)/("d"x) - "b" ("d"y)/("d"x))` = 4a
`("d"y)/("d"x) (y - "b") = (4"a")/2`
`("d"y)/("d"x) (y - "b")` = 2a
`y ("d"y)/("d"x) - 2"a" = "b" ("d"y)/("d"x)`
∵ `("d"y)/("d"x)` = y'
∴ yy' – 2a = by' .......(2)
Differentiating equation (2) with respect to ‘x’, we get
yy”+ y’y’ = by”
yy” + y’2 = by” ……. (3)
Substituting the b value in (3), we get
yy'' + (y')2 = `((yy"'" - 2"a")/(y"'"))y"''"`
`yy"''" + (y"'")^2 - y"''" ((yy"''" - 2"a")/(y"'"))` = 0
`yy"''" + (y"'")^2 - (yy"''"y"'")/y + (2"a"y"''")/(y"'")` = 0
`yy"''" + (y"'")^2 - yy"''" + (2"a"y"''")/(y"'")` = 0
`(y"'")^2 + (2"a"y"''")/(y"'")` = 0
Multiply by y', we get
`(y"'")^3 + (2"a"y"''" xx y"'")/(y"'")` = 0
(y')3 + 2ay'' = 0
Which is a required differential equation.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
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.
Reduce the following differential equation to the variable separable form and hence solve:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
Choose the correct option from the given alternatives:
The differential equation of all circles having their centres on the line y = 5 and touching the X-axis 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:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Solve the following differential equation:
x dy = (x + y + 1) dx
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation of the curve represented by xy = aex + be–x + x2
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
The differential equation for a2y = log x + b, is ______.
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.
