Some solutions for Homework 4
3.6: 12. In order to write down an ODE for this chain, consider both currents, i1 and i2 , indicated in figure 3.37. By Kirchhoff's law we need to add up all the voltages around a loop. We choose two loops: the left one and
Some solutions for Homework 3
3.6: 6. The answer is -6 + 6e5t - 30t - 75t2 - 125t3 . 2750 3.6: 10. We have Y (s) = L-1 Next, 10 + s F (s) + . (s + 4)(s + 6) (s + 4)(s + 6)
1 1 1 = - e-6t + e-4t . (s + 4)(s + 6) 2 2 10 + s 3 2 = - , (s + 4)(s + 6) s+4 s+6
Homework 2- Solutions.
1. f (x) = 1 A B C 1 = = + + . x(x2 - x - 6) x(x + 2)(x - 3) x x+2 x-3
Equations for the coefficients: x2 x 1 (A + B + C) = 0, (-A - 3B + 2C) = 0, (-6A) = 1.
Solve: A = -1/6, B = 1/10, C = 1/15. Have the indefinite integral, - 1 1 1
Solutions for Quiz on March 29
We need to diagonalize the matrix, 1 0 0 A = -1 0 1 . 1 1 0 First we find the eigenvalues. We need to solve 1- 0 0 -1 0 - 1 = 0. 1 1 0- We expand the determinant in cofactors of the first row. This row has only one nonzero e
Solution for Quiz 3
Find First shifting theorem: L-1 [F (s - a)] = eat L-1 [F (s)] . Here a = -1, and F (s - a) = 1/(s + 1)3 , so that F (s) = 1/s3 . We have L-1 [1/s3 ] = t2 /2. Therefore, L-1 Check that e-t 2 1 = t. (s + 1)3 2 d L[f (t)] ds L-1 1 . (s
Solution for Quiz 2
Find the Laplace transform of the function f (t) given by f (t) = 0, 0 t < 8, 5t, t 8.
First we express this function in terms of the Heaviside function, f (t) = 5tH(t - 8). Then we evaluate the Laplace transform: L[f ] = L[5tH(t - 8)]
Practice problems for Midterm 2
1. Suppose that A is a 100 100 matrix with all elements equal to zero except for the diagonal elements, aii = 3. (a) Calculate |A|. |A| = 3100 , because B is a diagonal matrix. (b) Suppose that the matrix B is the same a A,
Practice problems for Midterm 2
1. Suppose that A is a 100 100 matrix with all elements equal to zero except for the diagonal elements, aii = 3. (a) Calculate |A|. (b) Suppose that the matrix B is the same a A, except is has one more nonzero element: b13,
Math 421 Midterm #1 Solutions
1. Function f (t) is defined as follows: f (t) = t + 1, -t e ,
0,
t < 1, 1 t 2, t > 2.
(a) Sketch the function f (t).
(b) Express f (t) in terms of the Heaviside functions. f (t) = [H(t - 1) - H(t - 2)](t + 1) + H(t - 2)e-t
Sample problems for exam #1 in Math 421(1)
The exam will cover Laplace transform and Sections 5.4, 5.5, 6.1. On the exam, you will be expected to show all the steps. A bare asnwer is not sufficient. The exam will be shorter than this set of problems! Some
Homework 3- due Mon Feb 9.
1. Section 3.4, Problem 6. 2. Section 3.4, Problem 10. 3. Section 3.4, Problem 18. 4. Section 3.5, Problem 1. 5. Section 3.5, Problem 2. In all problems, please show all steps.
1
Homework 2 - due Mon Feb 2.
1. Suppose that the function f (x) is given by f (x) = x(x2 1 . - x - 6)
Evaluate f (x) dx by using partial fractions. Show your steps. Then write down the definite integral, 12 f (x) dx. 2. Section 3.2, Problem 7. 3. Section 3
Homework 1 - due Mon Jan 26.
1. Let u(x, y) = ey x . Find 5u , x5 Show all steps. 2. Calculate
0 A
2
2u . y 2
te-bt dt.
Show all steps. 3. Find det 4. Solve the initial value problem, y - 4y + 5y = 0, Show your work. y(0) = 1, y (0) = 2. 1 A 2 B
.
1
Solutions for practice problems for the Final, part 3
Note: Practice problems for the Final Exam, part 1 and part 2 are the same as Practice problems for Midterm 1 and Midterm 2. 1. Calculate Fourier Series for the function f (x), defined on [-2, 2], wher
Practice problems for the Final, part 3
Note: Practice problems for the Final Exam, part 1 and part 2 are the same as Practice problems for Midterm 1 and Midterm 2. 1. Calculate Fourier Series for the function f (x), defined on [-2, 2], where f (x) = -1,
Answers to some problems for the 1st midterm problem set
8. Answer: y1 (t) y2 (t) = = 1 - (3 - 2et-3 )H(t - 3), 30(1 - et-3 )H(t - 3).
(1)
9. Taking the Laplace, get (s2 - 3s + 2)Y (s) = Y (s) = Partial fractions, s2 (s 4 + s3 - 4s2 4 +s-4= . 2 s s2
s3 -