Find partial derivatives  f_x and  f_y for the function:

f(x,y) = 5x⁴y² + 6x² y³ 

at the point ,(1,-1)

Using Charpit’s method, find the complete integral of the differential equation:

p²x + q²y = z 

Find the complete solution of the equation px + qy = pq.

Show that the following function f is differentiable at (0,0):

Show that for the function f given by:

Find the general solution of the equation:

x(y²-z²)u_x + y(z²-x²)u_y +z(x²- y²)u₂ = 0

Solve:

Verify that the differential equation:

z(x²-y²-z^²)dx+(x+z)xzdy

                                  + x(z²-x2-xy)dz = 0

is integrable and find its integral.

If:

Z= x²y+2xy²

where x = sin zt

and

y = cost

find dz/dt  when t=0by (i) chain rule and by (ii) the direct substitution. 

By using the method of variation of parameter, find the general solution of the differential equation:

Solve the simultaneous equations:

Solve the differential equations:

Verigy Lagrange's mean value theorem for the function f defined by

f(x) = 2x²-7x-10over [2, 5].

a) If , show that:

(m+1)I_^m,n = x^m+l (log x)" -nI_m, n-1.

Hence find the value of fx²(logx)³dx.

Evaluate:

Solve the equation:

x² +15x³ +70x²+120x+64=0

given that its roots are in G.P.

Find the largest subset of R on which the function f: R→R defined as:

is continuous.

If y = e™ sin-¹ x, then show that (1-x²)y₂-xy, -m²y = 0. Hence using Leibnitz's formula, find the value of (1-x²)y_n+2-(2n+1)xy _n+1

Find the condition for the curves, ax² + by² =1 and a'x² + b'y² = 1 intersecting orthogonally.

Find the length of the cycloid and show that the line  divides it in the ratio 1: 3.