Differential Equations
4th Edition
ISBN: 9780495561989
Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall
Publisher: Cengage Learning
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Chapter 2.6, Problem 4E
To determine
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Example: Calculate the value of (aw/@x) at the point (x, y, z) = (2,-1,1) if:-
w = x2 + y2 + z?
z3 – xy + yz +y3 = 0
Assuming that x and y are the independent variables.
Write the given second order equation as its equivalent system of first order equations.
u" + 4.5u' – 7.5u = -7 sin(3t),
u(1) = -7.5,
u'(1) = -6
Use v to represent the "velocity function", i.e. v = u'(t).
Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.)
u' =
%3D
Now write the system using matrices:
u
d
+
dt
V
V
and the initial value for the vector valued function is:
u(1)
v(1)
Find two linearly independent solutions of 2x2y"-xy'+(-6x+1)y=0, x>0 of the form
Chapter 2 Solutions
Differential Equations
Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Prob. 5ECh. 2.1 - Prob. 6ECh. 2.1 - Consider the predator-prey system...Ch. 2.1 - Consider the predator-prey system dRdt=2R(1R...Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...
Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Prob. 12ECh. 2.1 - Prob. 13ECh. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Prob. 15ECh. 2.1 - Consider the system of predator-prey equations...Ch. 2.1 - Pesticides that kill all insect species are not...Ch. 2.1 - Some predator species seldom capture healthy adult...Ch. 2.1 - Prob. 19ECh. 2.1 - Consider the initial-value problem d2ydt2+kmy=0...Ch. 2.1 - A mass weighing 12 pounds stretches a spring 3...Ch. 2.1 - A mass weighing 4 pounds stretches a spring 4...Ch. 2.1 - Do the springs in an “extra firm’ mattress have a...Ch. 2.1 - Consider a vertical mass-spring system as shown in...Ch. 2.1 - Exercises 25—30 refer to a situation in which...Ch. 2.1 - Prob. 26ECh. 2.1 - Prob. 27ECh. 2.1 - Prob. 28ECh. 2.1 - Prob. 29ECh. 2.1 - Exercises 25—30 refer to a situation in which...Ch. 2.2 - Prob. 1ECh. 2.2 - Prob. 2ECh. 2.2 - Prob. 3ECh. 2.2 - Prob. 4ECh. 2.2 - Prob. 5ECh. 2.2 - Prob. 6ECh. 2.2 - Prob. 7ECh. 2.2 - Convert the second-order differential equation 1...Ch. 2.2 - Prob. 9ECh. 2.2 - Consider the system dxdt=2x+ydydt=2y and its...Ch. 2.2 - Eight systems of differential equations and four...Ch. 2.2 - Consider the modified predator-prey system...Ch. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 14ECh. 2.2 - Prob. 15ECh. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 17ECh. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 19ECh. 2.2 - Prob. 20ECh. 2.2 - Consider the four solution curves in the phase...Ch. 2.2 - Prob. 22ECh. 2.2 - Prob. 23ECh. 2.2 - Prob. 24ECh. 2.2 - Prob. 25ECh. 2.2 - Prob. 26ECh. 2.2 - Prob. 27ECh. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - Prob. 5ECh. 2.3 - In the damped harmonic oscillator, we assume that...Ch. 2.3 - Consider any damped harmonic oscillator equation...Ch. 2.3 - Consider any damped harmonic oscillator equation...Ch. 2.3 - In Exercises 9 and 10, we consider a mass sliding...Ch. 2.3 - In Exercises 9 and 10, we consider a mass sliding...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 6ECh. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 8ECh. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 11ECh. 2.4 - Prob. 12ECh. 2.4 - Consider the partially decoupled system...Ch. 2.5 - Prob. 1ECh. 2.5 - Prob. 2ECh. 2.5 - Prob. 3ECh. 2.5 - In Exercises 3—6, a system, an initial condition,...Ch. 2.5 - Prob. 5ECh. 2.5 - Prob. 6ECh. 2.5 - Using a computer or calculator, apply Euler’s...Ch. 2.5 - Prob. 8ECh. 2.6 - Consider the system dxdt=x+ydydt=y (a) Show that...Ch. 2.6 - Prob. 2ECh. 2.6 - Prob. 3ECh. 2.6 - Prob. 4ECh. 2.6 - Prob. 5ECh. 2.6 - Prob. 6ECh. 2.6 - Prob. 7ECh. 2.6 - (a) Suppose Y1(t) is a solution of an autonomous...Ch. 2.6 - Prob. 9ECh. 2.6 - Consider the system dxdt=2dydt=y2 (a) Calculate...Ch. 2.6 - Consider the system dxdt=2dydt=y2 Show that, for...Ch. 2.7 - Prob. 1ECh. 2.7 - In the SIR model, we assume that everyone in the...Ch. 2.7 - Vaccines make it possible to prevent epidemics....Ch. 2.7 - Prob. 4ECh. 2.7 - Prob. 5ECh. 2.7 - One of the basic assumptions of the SIR model is...Ch. 2.7 - Prob. 7ECh. 2.7 - Prob. 8ECh. 2.7 - Prob. 9ECh. 2.7 - Using =1.66 and the value of that you determined...Ch. 2.8 - Prob. 1ECh. 2.8 - Prob. 2ECh. 2.8 - Prob. 3ECh. 2.8 - Prob. 4ECh. 2.8 - Prob. 5ECh. 2 - Prob. 1RECh. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Prob. 7RECh. 2 - Prob. 8RECh. 2 - Prob. 9RECh. 2 - Prob. 10RECh. 2 - Prob. 11RECh. 2 - Prob. 12RECh. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Prob. 14RECh. 2 - Prob. 15RECh. 2 - Prob. 16RECh. 2 - Prob. 17RECh. 2 - Prob. 18RECh. 2 - Prob. 19RECh. 2 - Prob. 20RECh. 2 - Prob. 21RECh. 2 - Prob. 22RECh. 2 - Prob. 23RECh. 2 - Prob. 24RECh. 2 - Prob. 25RECh. 2 - Prob. 26RECh. 2 - Prob. 27RECh. 2 - Prob. 28RECh. 2 - Prob. 29RECh. 2 - Prob. 30RECh. 2 - In Exercises 31-34, a solution curve in the...Ch. 2 - Prob. 32RECh. 2 - Prob. 33RECh. 2 - Prob. 34RECh. 2 - Consider the partially decoupled system...Ch. 2 - Consider the partially decoupled system...Ch. 2 - Prob. 37RE
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- Write the equation „dr te +x dt dr in the form of +p(t)x=q(t). Then identify p(t) and q(t). dt Select one: O a. da - dt P() =-! 9(t)=e 1 t O b. dx dt p(t)=- 1 q(t)=e O c. da_ x dt t p(t)=- q(t)=ei +arrow_forwardConsider the equation system: y2 + w2 − 2xz = 0 x3 + y3 − z3 + w3 = 0 Prove that exists surrounding neighbourhood of point P = (1, −1, 1, 1) in which the solution (x, y, z, w) can be written with x and z as functions of (y, w)arrow_forwardts) Find two linearly independent solutions of 2x²y" - xy + (2x + 1)y=0, x > 0 of the form Y₁ = x¹(1+ a₁ + a₂c² + a3x³ + ...) Y₂ = x²(1 + b₁c + b₂x² + b3x³ + ...) where T1 T2. Enter T1 T2 a1 = a₂ = a3 = b₁ b₂ TE b3 N - || || || || =arrow_forward
- Write ay′′+ by′+ cy? = f(x) as a first order system of ODEsarrow_forwardConsider a normal, homogeneous, constant coefficient system of differential equations: yi = ay1 +A12Y2 +arrow_forwardY₁ = 1+ a3x³ + A6㺠+... Y₂ = x + b₁x¹ +b7x7 +... Enter the first few coefficients: Az = a6 = b₁ b7 ) Find two linearly independent solutions of y" + 6xy = = - 0 of the formarrow_forward
- D 8) Use the method of Variation of Parameters to solve y""-y' = t. calculate determinante!arrow_forwardEliminate the Arbitrary Constant: y = c1x² + c2e¬x a. 2(x + 1)y' – (x² – 2)y" – x(x + 2)y = 0 b. 2(х + 1)у — (x2 — 2)у" — х(х + 2)у' %3D 0 с. 2(х + 1)у — (х2 — 2)у' — x(х + 2)у" %3D0 d. 2(x + 1)y" – (x² – 2)y – x(x + 2)y" = 0 e. none of thesearrow_forwardPlease SHOW ALL WORKarrow_forward
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