Differential Calculus

1. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)dy=0
2. y(9x-2y)dx-x(6x-y)dy=0; when: x=1; y=1/2

Hi what is the clear process of this
Thanks for your help.

lim 1-cos x / x^2
->0

ans.= 0.5

A reservoir is in the form of a frustum of a cone with upper base of radius 9 ft and lower base of radius 4 ft and altitude of 10 ft. The water in the reservoir is x ft deep. If the level of the water is increasing at 4 ft/min, how fast is the volume of the water in the reservoir increasing when its depth is 2 ft ?

How can I laplace transform this (t sin(at) cos(bt))²

z1= 4^(e^[(-i)(π/4)])
z2= -5i
z3= 3√3<30°

Note: < is the angle sign. I don't have symbol for.the angle sign in my keyboard.

Solve for:
[z1^(4cos z4)]/log(sub z1)z3

Please help :(

Please help!!! :( I need it before midnight strikes. It's for my hhomework but I'm having a hard time. Thank you.

1. For a parallel RLC circuit, R = 1/3 ohms, C = 0.5 farads, and L = 2/13 henrys. If the
current source of the circuit produces i(t) = 20e^(–3t)sin 2t, find the voltage v at any time t
if v(0) = 2 and v’(0) = 1.5.

2. For a series RLC circuit, R = 4 ohms, C = 0.25 farads, and L = 2 henrys. If the voltage
source of the circuit produces v(t) = 6e
–tcos t, find the current i at any time t if i(0) = 4
and i’(0) = 1.
�

(D^6 – 10D^5 + 20D^4 + 50D^3 – 121D^2 – 40D + 100)y = (64x^3)e
^(5x)
(Use Exponential Shift)�

xy' = y + 2 √xy

2xy' + y³ e^(-2x) = 2xy

(x²+4) y' + 3 xy = x