Solving Differential Equations, Volumes of Solids and Taylor Series

See attached file for full problem description.

1. First find a general solution of the differential equation dy 3y
dx
= . Then
find a particular solution that satisfies the initial condition that
y(1) = 4.
2. Solve the initial value problem dy y3
dx
= , y(0) = 1.
3. Nyobia had a population of 3 million in 1985. Assume that this
country's population is growing continuously at a 5% annual rate and
that Nyobia absorbs 40,000 newcomers per year. What will its
population be in the year 2015?
4. Find the center and radius of the circle described in the equation
2x2 + 2y2 - 6x + 2y = 3.
5. Find an equation of the ellipse with center (-2, 1), horizontal major axis
10, and eccentricity 2
5
.
6. Determine whether or not the sequence n 2n 3
na = 2 + converges and find
its limit if it does converge.
7. Write the Taylor series with center zero for the function
f (x)=ln(1+x2).
8. Given a = 2i + 3j, b = 3i + 5j, and c = 8i + 11j express c in the form ra +
sb where r and s are scalars.
9. Given a = <4, -3, -1> and b = <1, 4, 6>, find a × b.
10. Find the arc length of the curve given by x = cos 3t, y = sin 3t,
z = 4t, from t = 0 to t =
2
π .
11. Compute the first-order partial derivatives of f (x,y) 2x
x y
=
−
.
12. Use the method of Lagrange multipliers to find the extreme values of
3x - 4y + 12z on the spherical surface with equation x2 + y2 + z2 = 1.
13. Evaluate
3 1
1 0
(2x 3y)∫ ∫ − dxdy .
14. Find the volume of the solid bounded by x = 0, y = 0, z = 0, and
x + 2y + 3z = 6 by triple integration.
15. Calculate the divergence and curl of the vector field
F(x, y, z) = 2xi + 3yj + 4zk.
16. Use Green's theorem to evaluate
C
��∫ Pdx +Qdy
P(x, y) = xy, Q(x, y) = ex; C is the curve that goes from (0, 0) to (2, 0)
along the x-axis and then returns to (0, 0) along the parabola y = 2x - x2.

Solving Differential Equations, Volumes of Solids and Taylor Series are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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1 Write the Taylorseries with center zero for the function f(x) = ln(l+x^2).
2 Given a =2i +3j, b = 3i +5j , and c = 8i + 11j express c in the form ra+ sb where r and s are scalars.

The starting point for this problem is the representation of the natural exponential function by the corresponding Taylorseries in powers of x:
e^x= 1 + x +(1/2!)x^2 + (1/3!)x^3 + ......(1/n!)x^n + ........, x E R
a) Let f(t) = e^(-t^2) , t E R. Define the Taylorseries for f from the Taylorseries for e^x.
b) Let erf(x)=

This problems is from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution.
Problem:
Obtain the Taylorseries ... (see attachment) for the function ... (see attachment)

A 20 ohm resistor, a .05 farad capacitor, and an alternating power source equal to 40 cos(t) are placed in series. If the initial charge on the capacitor is 3 coulombs, find a general formula for the charge at any time t.

Using the fact that 1+x = 4+(x-3), find the Taylorseries about 3 for g. Give explicitly the numbers of terms. When g(x)=square root of 1+x
Check the first four terms in the Taylorseries above and use these to find cubic Taylor polynomials about 3 for g.
Use multiplication of Taylorseries to find the quartic Taylor polyn