Advertisements
Advertisements
प्रश्न
If y = Aemx + Benx, show that y2 – (m + n)y1 + mny = 0.
Advertisements
उत्तर
y = Aemx + Benx
Differentiating w.r.t. x, we get
`"dy"/"dx" = "A""d"/"dx"(e^(mx)) + "B""d"/"dx"(e^(nx))`
= `"Ae"^(mx)."d"/"dx"(mx) + "Be"^(nx)."d"/"dx"(nx)`
= Aemx . m + Benx . n
= y1 = mAemx + nBenx ...(1)
Differentiating again w.r.t. x, we get
y2 = `m"A""d"/"dx"(e^(mx)) + n"B""d"/"dx"(e^(nx))`
= `m"Ae"^(mx)."d"/"dx"(mx) + n"Be"^(nx)."d"/"dx"(nx)`
= mAemx . m + nBenx . n
∴ y2 = m2Aemx + n2Benx ...(2)
∴ y2 – (m + n)y1 + mny = (m2Aemx + n2Benx) – (m + n)(mAemx + nBenx) + mn(Aemx + Benx) ...[By (1), (2)]
= m2Aemx + n2Benx – m2Aemx – mnBemx – n2Benx + mnAemx + mnBenx
= 0
∴ y2 – (m + n)y1 + mny = 0
APPEARS IN
संबंधित प्रश्न
If y=eax ,show that `xdy/dx=ylogy`
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
if `x^y + y^x = a^b`then Find `dy/dx`
Is |sin x| differentiable? What about cos |x|?
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `"dy"/"dx"` ; if x = sin3θ , y = cos3θ
Find `"dy"/"dx"` ; if y = cos-1 `("2x" sqrt (1 - "x"^2))`
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
If ex + ey = ex+y, then show that `"dy"/"dx" = -e^(y - x)`.
Find `"dy"/"dx"`, if : x = a(1 – cosθ), y = b(θ – sinθ)
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Differentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`
Differentiate xx w.r.t. xsix.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)
Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.
Find the nth derivative of the following : cos x
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : cos (3 – 2x)
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
If `xsqrt(1 - y^2) + ysqrt(1 - x^2)` = 1, then show that `"dy"/"dx" = -sqrt((1 - y^2)/(1 - x^2)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
If y2 = a2cos2x + b2sin2x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, yex + xey = 1
Find `"dy"/"dx"` if, xy = log (xy)
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
Choose the correct alternative.
If `"x"^4."y"^5 = ("x + y")^("m + 1")` then `"dy"/"dx" = "y"/"x"` then m = ?
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
`(dy)/(dx)` of `2x + 3y = sin x` is:-
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0
Find `(d^2y)/(dy^2)`, if y = e4x
Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.
Find `dy/dx` if , x = `e^(3t), y = e^(sqrtt)`
Find `dy/dx` if, x = `e^(3t)`, y = `e^sqrtt`
Find `dy/dx if , x = e^(3t) , y = e^sqrtt`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
If log(x + y) = log(xy) + a, then show that `dy/dx = (-y^2)/x^2`
