Advertisements
Advertisements
Question
If ax2 + 2hxy + by2 = 0, then show that `("d"^2"y")/"dx"^2` = 0
Advertisements
Solution
ax2 + 2hxy + by2 = 0 ....(i)
Differentiating both sides w.r.t. x, we get
`"a"("2x") + "2h" * "d"/"dx" ("xy") + "b"("2y") "dy"/"dx" = 0`
∴ `2"ax" + 2"h" ["x" * "dy"/"dx" + "y"(1)] + 2"by" "dy"/"dx" = 0`
∴ `2"ax" + 2"hx" "dy"/"dx" + 2"hy" + 2"by" "dy"/"dx" = 0`
∴ `2 "dy"/"dx" ("hx" + "by") = - 2"ax" - 2"hy"`
∴ `2 "dy"/"dx" = (-2("ax" + "hy"))/("hx" + "by")`
∴ `"dy"/"dx" = (- ("ax" + "hy"))/("hx" + "by")` ....(i)
ax2 + 2hxy + by2 = 0
∴ ax2 + hxy + hxy + by2 = 0
∴ x(ax + hy) + y(hx + by) = 0
∴ y(hx + by) = - x(ax + hy)
∴ `"y"/"x" = (- ("ax" + "hy"))/("hx" + "by")` ...(ii)
From (i) and (ii), we get
`"dy"/"dx" = "y"/"x"` ....(iii)
Again, differentiating both sides w.r.t. x, we get
`("d"^2"y")/"dx"^2 = ("x" * "dy"/"dx" - "y" * "d"/"dx" ("x"))/"x"^2`
`= ("x" * ("y"/"x") - "y"(1))/"x"^2` ....[From (iii)]
`= ("y - y")/"x"^2`
`= 0/"x"^2`
∴ `("d"^2"y")/"dx"^2 = 0`
APPEARS IN
RELATED QUESTIONS
If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`
Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`
Find the second order derivative of the function.
x20
Find the second order derivative of the function.
x . cos x
Find the second order derivative of the function.
sin (log x)
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If ey (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.
If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2
If x7 . y9 = (x + y)16 then show that `"dy"/"dx" = "y"/"x"`
Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
`sin xy + x/y` = x2 – y
(x2 + y2)2 = xy
If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:
`"Find" (d^2y)/(dx^2) "if" y=e^((2x+1))`
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
Higher order derivatives are obtained by which process?
If \[y=\sin^{-1}x\], what is \[\frac{dy}{dx}\]?
After differentiating \[\sqrt{1-x^2}\cdot\frac{dy}{dx}=1\], which equation results?
Which equation follows from \[(1-x^2)\cdot2y_{1}y_{2}+y_{1}^{2}(0-2x)=0\]?
