Explanation Consider the given systems of differential equation as { y ′ 1 = 4 y 2 y ′ 2 = 4 y 1 + 32 t 2 . The coefficient matrix of the corresponding systems of differential equations is A = [ 0 4 4 0 ] . Note that, a single vector equation is y ′ = A y + g ( t ) , where A is coefficient matrix and g ( t ) = [ 0 32 t 2 ] . Hence, a single vector equation is y ′ = [ 0 4 4 0 ] y + [ 0 32 t 2 ] , where y = [ y 1 y 2 ] . Characteristic equation of the coefficient matrix A = [ 0 4 4 0 ] is det ( A − λ I ) = 0 , where λ eigenvalue and I is 2 × 2 identity matrix . Simplify for the det ( A − λ I ) = 0 using the coefficient matrix as follows. det ( A − λ I ) = 0 det ( [ 0 4 4 0 ] − λ [ 1 0 0 1 ] ) = 0 det ( [ − λ 4 4 − λ ] ) = 0 λ 2 − 16 = 0 λ = ± 4 Compute the eigenvector t ( 1 ) = [ t 1 t 2 ] corresponding to the eigenvalue λ = 4 as follows. A t − ( 4 ) t = 0 [ − 4 4 4 − 4 ] [ t 1 t 2 ] = [ 0 0 ] [ − 4 t 1 + 4 t 2 4 t 1 − 4 t 2 ] = [ 0 0 ] − 4 t 1 + 4 t 2 = 0 and 4 t 1 − 4 t 2 = 0 t 1 1 = t 2 1 Hence, the possible vector corresponding to the eigenvalue λ = 4 is t ( 1 ) = [ 1 1 ] . Compute the eigenvector t ( 2 ) = [ t 1 t 2 ] corresponding to the eigenvalue λ = − 4 as follows. A t − ( − 4 ) t = 0 [ 4 4 4 4 ] [ t 1 t 2 ] = [ 0 0 ] [ 4 t 1 + 4 t 2 4 t 1 + 4 t 2 ] = [ 0 0 ] 4 t 1 + 4 t 2 = 0 and 4 t 1 + 4 t 2 = 0 t 1 1 = t 2 − 1 Hence, the possible vector corresponding to the eigenvalue λ = − 4 is t ( 2 ) = [ 1 − 1 ] . Thus, the homogenous solution of the system of differential equation is y h = c 1 e 4 t [ 1 1 ] + c 2 e − 4 t [ 1 − 1 ] . Now obtain the particular solution of the system of differential equation using undetermined coefficient as follows. Let the particular solution of the differential equation be of the form y p = [ a 1 b 1 ] t 2 + [ a 2 b 2 ] t + [ a 3 b 3 ] Differentiate the solution with respect to t . y ′ p = [ a 1 b 1 ] d d t ( t 2 ) + [ a 2 b 2 ] d d t ( t ) + [ a 3 b 3 ] d d t ( 1 ) = 2 [ a 1 b 1 ] t + [ a 2 b 2 ] ( 1 ) + [ a 3 b 3 ] ( 0 ) = [ 2 a 1 2 b 1 ] t + [ a 2 b 2 ] Substitute y p = [ a 1 b 1 ] t 2 + [ a 2 b 2 ] t + [ a 3 b 3 ] in the equation y ′ = [ 0 4 4 0 ] y + [ 0 32 t 2 ]

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 4, Problem 21RQ

To determine

The general solution of the given systems of differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4, Problem 20RQ

Chapter 4, Problem 22RQ

Students have asked these similar questions

Solve the linear system of equations attached using Gaussian elimination (not Gauss-Jordan) and back subsitution. Remember that: A matrix is in row echelon form if Any row that consists only of zeros is at the bottom of the matrix. The first non-zero entry in each other row is 1. This entry is called aleading 1. The leading 1 of each row, after the first row, lies to the right of the leading 1 of the previous row.

7. Show that for R sufficiently large, the polynomial P(z) in Example 3, Sec. 5, satisfies the inequality |P(z)| R. Suggestion: Observe that there is a positive number R such that the modulus of each quotient in inequality (9), Sec. 5, is less than |an|/n when |z| > R.

9. Establish the identity 1- 1+z+z² + 2n+1 ... +z" = 1- z (z1) and then use it to derive Lagrange's trigonometric identity: 1 1+ cos cos 20 +... + cos no = + 2 sin[(2n+1)0/2] 2 sin(0/2) (0 < 0 < 2л). Suggestion: As for the first identity, write S = 1+z+z² +...+z" and consider the difference S - zS. To derive the second identity, write z = eie in the first one.

Chapter 4 Solutions

Advanced Engineering Mathematics

Ch. 4.1 - Prob. 1P Ch. 4.1 - Prob. 2P Ch. 4.1 - Prob. 3P Ch. 4.1 - Prob. 4P Ch. 4.1 - If you extend Example 1 by a tank T3 of the same...Ch. 4.1 - Find a “general solution” of the system in Prob....Ch. 4.1 - In Example 2 find the currents: 7. If the initial...Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Find a general solution of the given ODE (a) by...

Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Prob. 14P Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - Find a real general solution of the following...Ch. 4.3 - Prob. 9P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - 10–15 IVPs Solve the following initial value...Ch. 4.3 - Prob. 12P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Prob. 16P Ch. 4.3 - Prob. 17P Ch. 4.3 - Prob. 18P Ch. 4.3 - Prob. 19P Ch. 4.4 - Prob. 1P Ch. 4.4 - Prob. 2P Ch. 4.4 - Prob. 3P Ch. 4.4 - Prob. 4P Ch. 4.4 - Prob. 5P Ch. 4.4 - Prob. 6P Ch. 4.4 - Prob. 7P Ch. 4.4 - Prob. 8P Ch. 4.4 - Prob. 9P Ch. 4.4 - Prob. 10P Ch. 4.4 - Prob. 11P Ch. 4.4 - Prob. 12P Ch. 4.4 - Prob. 13P Ch. 4.4 - Prob. 14P Ch. 4.4 - Prob. 15P Ch. 4.4 - Prob. 16P Ch. 4.4 - Prob. 17P Ch. 4.5 - Prob. 1P Ch. 4.5 - Prob. 2P Ch. 4.5 - Prob. 4P Ch. 4.5 - Prob. 5P Ch. 4.5 - Prob. 6P Ch. 4.5 - Prob. 7P Ch. 4.5 - Prob. 8P Ch. 4.5 - Prob. 9P Ch. 4.5 - Prob. 10P Ch. 4.5 - Prob. 11P Ch. 4.5 - Prob. 12P Ch. 4.5 - Prob. 13P Ch. 4.6 - Prob. 1P Ch. 4.6 - Prob. 2P Ch. 4.6 - Prob. 3P Ch. 4.6 - Prob. 4P Ch. 4.6 - Prob. 5P Ch. 4.6 - Prob. 6P Ch. 4.6 - Prob. 7P Ch. 4.6 - Prob. 9P Ch. 4.6 - Prob. 10P Ch. 4.6 - Prob. 11P Ch. 4.6 - Prob. 12P Ch. 4.6 - Prob. 13P Ch. 4.6 - Prob. 14P Ch. 4.6 - Prob. 15P Ch. 4.6 - Prob. 16P Ch. 4.6 - Prob. 17P Ch. 4.6 - Prob. 19P Ch. 4 - Prob. 1RQ Ch. 4 - Prob. 2RQ Ch. 4 - How can you transform an ODE into a system of...Ch. 4 - What are qualitative methods for systems? Why are...Ch. 4 - Prob. 5RQ Ch. 4 - Prob. 6RQ Ch. 4 - What are eigenvalues? What role did they play in...Ch. 4 - Prob. 8RQ Ch. 4 - Prob. 9RQ Ch. 4 - Prob. 10RQ Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Prob. 15RQ Ch. 4 - Prob. 16RQ Ch. 4 - Prob. 17RQ Ch. 4 - Prob. 18RQ Ch. 4 - Prob. 19RQ Ch. 4 - Prob. 20RQ Ch. 4 - Prob. 21RQ Ch. 4 - Prob. 22RQ Ch. 4 - Prob. 23RQ Ch. 4 - Prob. 24RQ Ch. 4 - Prob. 25RQ Ch. 4 - Network. Find the currents in Fig. 103 when R = 1...Ch. 4 - Prob. 27RQ Ch. 4 - Prob. 28RQ Ch. 4 - Find the location and kind of all critical points...Ch. 4 - Find the location and kind of all critical points...

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.