The population of a town grows at a rate proportional to the population at any time. Its initial population of 500 increases by 15% in 10 years. What will be the population in 30 years?

State Euler’s theorem for homogeneous functions and verify it for the function:

Find partial derivatives  f_x and  f_y for the function:

f(x,y) = 5x⁴y² + 6x² y³ 

at the point ,(1,-1)

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 

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.

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

b) 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. 

Solve the simultaneous equations:

 Solve the differential equations:

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.

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.