Explanation Given: The given non-homogeneous system is, X ′ = ( 2 − 2 8 − 6 ) X + ( 1 3 ) e − 2 t t Calculation: The system is given in the form X ′ = A X + F ( t ) (1). Consider, the characteristic equation of a matrix A as | A − λ I | = 0 , where I is identity matrix. Now, the characteristic equation of the coefficient matrix A . is | ( A − λ I ) | = 0 | 2 − λ − 2 8 − 6 − λ | = 0 ( 2 − λ ) ( − 6 − λ ) + 16 = 0 − 12 − 2 λ + 6 λ + λ 2 + 16 = 0 Simplifying the above obtained expression it gives, λ 2 + 4 λ + 4 = 0 ( λ + 2 ) 2 = 0 λ = − 2 , − 2 So, Eigen values are λ 1 = − 2 , λ 2 = − 2 , . Now evaluating the Eigen vector for λ 1 = − 2 , ( A − ( − 2 ) I 0 ) = ( 4 − 2 0 8 − 4 0 ) On solving, it gives 4 K 1 − 2 K 2 = 0 which on further evaluation gives 2 K 1 = K 2 . So, the corresponding Eigen vector is K = ( 1 2 ) . Thus, the value of corresponding solution vector is X 1 = ( 1 − 1 ) e t . There are two repeated Eigen value, so consider another Eigen vector is P = ( a b ) . Now, solve the next system ( A − 2 I ) P = K to get the another Eigen vector. ( A − ( − 2 ) I ) ( a b ) = K ( 4 − 2 8 − 4 ) ( a b ) = ( 1 2 ) On solving, it gives 4 a − 2 b = 1 In the above obtained expression, there is only one equation and two variables, so for satisfying the above expression consider a = 1 2 , then b becomes 1 2 . So the corresponding Eigen vector is P = ( 1 2 1 2 ) . Thus, the value of corresponding solution vector X 2 is, X 2 = K t e λ 1 t + P e λ 2 t = ( 1 2 ) t e − 2 t + ( 1 2 1 2 ) e − 2 t . Consider the coefficient matrix A having real Eigen values of homogeneous system, K be an Eigen vector and X 1 , X 2 be the solution vectors of the corresponding Eigen values then, the value of complementary function X c is, X c = C 1 X 1 + C 2 X 2 Substituting the value of X 1 and X 2 in above expression it gives, X c = C 1 ( 1 2 ) e − 2 t + C 2 [ ( 1 2 ) t e − 2 t + ( 1 2 1 2 ) e − 2 t ] (2) Now, the entries of X 1 form the first column of solution matrix ϕ ( t ) , and the entries of X 2 form the second column of ϕ ( t ) , which means the value of ϕ ( t ) is ϕ ( t ) = ( e − 2 t ( t + 1 2 ) e − 2 t 2 e − 2 t ( 2 t + 1 2 ) e − 2 t ) Evaluating determinant of the above matrix ϕ ( t ) , det [ ϕ ( t ) ] = | e − 2 t ( t + 1 2 ) e − 2 t 2 e − 2 t ( 2 t + 1 2 ) e − 2 t | = e − 2 t ⋅ ( 2 t + 1 2 ) e − 2 t − 2 e − 2 t ⋅ ( t + 1 2 ) e − 2 t = 2 t e − 4 t + 1 2 e − 4 t − 2 t e − 4 t − e − 4 t = − 1 2 e − 4 t Now, the value of the value of ϕ − 1 ( t ) is ϕ − 1 ( t ) = 1 | ϕ ( t ) | ⋅ ( Adj . ϕ ( t ) ) = 1 ( − 1 2 e − 4 t t ) ⋅ [ Adj

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 8.3, Problem 26E

To determine

To solve: The given non-homogeneous system by using variation of parameters.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 25E

Chapter 8.3, Problem 27E

Students have asked these similar questions

We consider a one-period market with the following properties: the current stock priceis S0 = 4. At time T = 1 year, the stock has either moved up to S1 = 8 (with probability0.7) or down towards S1 = 2 (with probability 0.3). We consider a call option on thisstock with maturity T = 1 and strike price K = 5. The interest rate on the money marketis 25% yearly.(a) Find the replicating portfolio (φ, ψ) corresponding to this call option.(b) Find the risk-neutral (no-arbitrage) price of this call option.(c) We now consider a put option with maturity T = 1 and strike price K = 3 onthe same market. Find the risk-neutral price of this put option. Reminder: A putoption gives you the right to sell the stock for the strike price K.1(d) An investor with initial capital X0 = 0 wants to invest on this market. He buysα shares of the stock (or sells them if α is negative) and buys β call options (orsells them is β is negative). He invests the cash balance on the money market (orborrows if the amount is…

Determine if the two statements are equivalent using a truth table

Use Pascal's triangle to expand the binomial (6m+2)^2

Chapter 8 Solutions

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

Knowledge Booster

Learn more about

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

The given non-homogeneous system by using variation of parameters.

Solution Summary: The author explains how to solve the given non-homogeneous system by using variation of parameters.

Concept explainers

Videos

To solve: The given non-homogeneous system by using variation of parameters.

Want to see the full answer?

Chapter 8 Solutions