 Chapter 4.4.1: Find out without calculation whether doubling the flow rate in Exam...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: (General solution) Prove that (2) includes every solution of (I).
 Chapter 4.1: State some applications that can be modeled by systems of ODEs.
 Chapter 4.4.1: What happens in Example 1 if we replace T2 by a tank containing 500...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.2: What is population dynamics? Give examples.
 Chapter 4.4.1: Derive the eigenvectors III Example 1 without consulting this book.
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.3: How can you transform an ODE into a system of ODEs?
 Chapter 4.4.1: In Example 1 find a "general solution" for any ratio a = (flow rate...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.4: What are qualitative methods for systems? Why are they important?
 Chapter 4.4.1: If you extend Example I by a tank T3 of the same size as the others...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.5: What is the phase plane? The phase plane method? The phase portrait...
 Chapter 4.4.1: Find a "general solution" of the system in Prob. 5.
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.6: What is a critical point of a system of ODEs? How did we classify t...
 Chapter 4.4.1: Find the currents in Example 2 if the initial cunents are 0 and  ...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.7: What are eigenvalues? What role did they play in this chapter?
 Chapter 4.4.1: Find the cunents in Example 2 if the resistance of R1 and R2 is do...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.8: What does stability mean in general? In connection with critical po...
 Chapter 4.4.1: What are the limits of the CUlTents in Example 27 Explain them in t...
 Chapter 4.4.3: Find a real general solutiun of the following systems. (Show the de...
 Chapter 4.4.4: Determine the type and stability of the critical point. Then find a...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Find a general solution. (Show the details of your work.)
 Chapter 4.9: What does linearization of a system mean? Give an example.
 Chapter 4.4.1: Find the cunems in Example 2 if the capacitance is changed to C = ...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: What kind of curves are the trajectories of the following ODEs in t...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: CAS EXPERIMENT. Undetermined Coefficients. Find out experimentally ...
 Chapter 4.10: What is a limit cycle? When may it occur in mechanics?
 Chapter 4.4.1: Find a general solution of the given ODE (a) by first converting it...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: What kind of curves are the trajectories of the following ODEs in t...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Solve {showing details): y~ = 2Y2 + 4ty~ = 2YI  2t)'1(0) = 4, )'2...
 Chapter 4.11: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.1: Find a general solution of the given ODE (a) by first converting it...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: What kind of curves are the trajectories of the following ODEs in t...
 Chapter 4.4.5: Determine the location and type of all critical points by lineariza...
 Chapter 4.4.6: Solve {showing details):
 Chapter 4.12: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.1: Find a general solution of the given ODE (a) by first converting it...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: (Damped oscillation) Solve y" + 4y' + 5y = O. What kind of curves d...
 Chapter 4.4.5: (Trajectories) What kind of curves are the trajectories of ,~y" + ...
 Chapter 4.4.6: Solve {showing details):Y; = YI + 2.\"2 + e2t  2ty~ =  )'2 + 1 + ...
 Chapter 4.13: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.1: Find a general solution of the given ODE (a) by first converting it...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: (Transformation of variable) What happens to the system (1) and its...
 Chapter 4.4.5: (Trajectories) Write the ODE y"  4y + )'3 = 0 as a system. ~llive ...
 Chapter 4.4.6: Solve {showing details):
 Chapter 4.14: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.1: Find a general solution of the given ODE (a) by first converting it...
 Chapter 4.4.3: Solve the following initial value problems. (Show the details.)
 Chapter 4.4.4: (Types of critical points) Discuss the critical points in (10)( 1...
 Chapter 4.4.5: (Trajectories) What is the radius of a real general solution of y" ...
 Chapter 4.4.6: Solve {showing details):
 Chapter 4.15: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.1: TEAM PROJECT. Two Masses on Springs. (a) Set up the model for the (...
 Chapter 4.4.3: Find a general solution by conversion to a single ODE.The system in...
 Chapter 4.4.4: (Perturbation of center) If a system has a center as its critical p...
 Chapter 4.4.5: (Trajectories) In Prob. 14 add a linear damping tenn to get y" + 2y...
 Chapter 4.4.6: Solve {showing details):
 Chapter 4.16: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.3: Find a general solution by conversion to a single ODE. The system i...
 Chapter 4.4.4: (Perturbation) The system in Example 4 in Sec. 4.3 has a center as ...
 Chapter 4.4.5: (Pendnlum) To what state (position, speed, direction of motion) do ...
 Chapter 4.4.6: (Network) Find the currents in Fig. 97 when R = 2.5 D. L = 1 H, C =...
 Chapter 4.17: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.3: (Mixing problem, Fig. 87) Each of the two tanks contains 400 gal of...
 Chapter 4.4.4: CAS EXPERIMENT. Phase Portraits. Graph phase portraits for the syst...
 Chapter 4.4.5: (Limit cycle) What is the essential difference between a limit cycl...
 Chapter 4.4.6: (Network) Find the currents in Fig. 97 when R = I D. L = 10 H, C = ...
 Chapter 4.18: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.3: (Network) Show that a model for the currents I] (1) and 12(t) in Fi...
 Chapter 4.4.4: WRITING EXPERIMENT. Stability. Stability concepts are basic in phys...
 Chapter 4.4.5: CAS EXPERIMENT. Deformation of Limit Cycle. Convert the van der Pol...
 Chapter 4.4.6: (Network) Find the CUiTents in Fig. 98 when R1 = 2 D, R2 = 8 n. L =...
 Chapter 4.19: Find a general solution. Determine the kind and stability of the cr...
 Chapter 4.4.3: CAS PROJECT. Phase Portraits. Graph some of the figures in this sec...
 Chapter 4.4.4: (Stability chart) Locate the critical points of the systems (0)(14...
 Chapter 4.4.5: TEAM PROJECT. Selfsustained oscillations. (a) Van der Pol Equation...
 Chapter 4.4.6: WRITING PROJECT. Undetermined Coefficients. Write a short report in...
 Chapter 4.20: Find a general solution. (Show the details.) y~ = 3)'2 + 6t y~ = 12...
 Chapter 4.21: Find a general solution. (Show the details.))'~ = )'1 + 2.\'2 + e2t
 Chapter 4.22: Find a general solution. (Show the details.)
 Chapter 4.23: Find a general solution. (Show the details.)
 Chapter 4.24: Find a general solution. (Show the details.) y~ = Y1  2Y2  sin ty...
 Chapter 4.25: Find a general solution. (Show the details.)
 Chapter 4.26: (Mixing problem) Tank Tl in Fig. 99 contains initially 200 gal of w...
 Chapter 4.27: (Critical point) What kind of critical point does y' = Ay have if A...
 Chapter 4.28: (Network) Find the currents in Fig. 100. where R1 = 0.5 fl, R2 = 0....
 Chapter 4.29: (Network) Find the currents in Fig. 10 1 when R = 10 fl, L = 1.25 H...
 Chapter 4.30: Detelmine the location and kind of all critical points of the given...
 Chapter 4.31: Detelmine the location and kind of all critical points of the given...
 Chapter 4.32:
 Chapter 4.33: Detelmine the location and kind of all critical points of thegiven ...
Solutions for Chapter Chapter 4: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 4
Get Full SolutionsSince 129 problems in chapter Chapter 4 have been answered, more than 46173 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 4 includes 129 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.