Engineering Mat...

The non-zero of n for which the differential equation (3xy² + n² x²y) dx + (nx³ + 3x²y) dy = 0, x ≠ 0 become exact differential equation is:

If u = log (x² + y²) where x + y + xy = 4, then \(\frac{{du}}{{dx}}\) at (1, 1) is

For the function f(x) = sin x - cos x, use Rolle’s Theorem and find the point where derivative vanishes in [0, π].

The value of ‘C’ of the Cauchy’s mean value theorem for f(x) = e^x and g(x) = e^-x in [2, 3] is _____.

If \(u = x\log xy\), where \({x^3} + {y^3} + 3xy = 1\;\)then \(\frac{{du}}{{dx}}\) is equal to

Find C of Cauchy’s mean value theorem for the function 1/x and 1/x² in [4, 6]

If the differential equation corresponding to the family of curves Y = (A + Bx) e^3x is given by

\(\frac{{{d^2}y}}{{d{x^2}}} = a\frac{{dy}}{{dx}} + by,\) then (a - b) is _____

The maximum value of function f(x, y) = x³ y² (1 - x - y) for x, y ϵ (0, ∞) is

Find the maximum and minimum values of f(x) = sin x + cos 2x where \(0\le x \le 2\pi\)

If \({y_1} = \frac{{{x_2} \cdot {x_3}}}{{{x_1}}},\;{y_2} = \frac{{{x_3}{x_1}}}{{{x_2}}},\;{y_3} = \frac{{{x_1}{x_2}}}{{{x_3}}},\) then the Jacobian of y₁, y₂, y₃ w.r.t. x₁, x₂, x₃ is ______