Differential Equations: An Introduction to Modern Methods and Applications
Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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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…
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Chapter 4.7, Problem 16P

In each of Problems 14 through 21, find the general solution of the given differential equation.

In Problems 20 and 21, g is an arbitrary continuous function.

y ″ + 4 y ' + 4 y =   t − 2 e − 2 t ,             t > 0

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Chapter 4 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - In Problems 1 through 7, determine whether the...Ch. 4.1 - A mass weighing stretches a spring . What is the...Ch. 4.1 - A mass attached to a vertical spring is slowly...Ch. 4.1 - A mass weighing stretches a spring . The mass is...
Ch. 4.1 - A mass of stretches a spring. The mass is set in...Ch. 4.1 - A mass weighing 3lb stretches a spring 3in. The...Ch. 4.1 - A series circuit has a capacitor of 0.25...Ch. 4.1 - A mass of stretches a spring . Suppose that the...Ch. 4.1 - A mass weighing 16lb stretches a spring 3in. The...Ch. 4.1 - A spring is stretched by a force of (N). A mass...Ch. 4.1 - A series circuit has a capacitor of 105farad, a...Ch. 4.1 - Suppose that a mass m slides without friction on a...Ch. 4.1 - Duffing’s Equation For the spring-mass system...Ch. 4.1 - A body of mass is attached between two springs...Ch. 4.1 - A cubic block of side and mass density per unit...Ch. 4.1 - In Problems through , we specift the mass, damping...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - In Problems through , we specift the mass, damping...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - In Problems 22 through 26, we specift the mass,...Ch. 4.1 - The Linear Versus the Nonlinear Pendulum. Convert...Ch. 4.1 - (a) Numerical simulations as well as intuition...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems 1 through 8, determine the...Ch. 4.2 - In each of the Problems through, determine the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - In each of the Problems 9 through 14, find the...Ch. 4.2 - In each of the Problems 9 through 14, find the...Ch. 4.2 - In each of the Problems through, find the...Ch. 4.2 - Verify that and are two solutions of the...Ch. 4.2 - Consider the differential operator T defined by...Ch. 4.2 - Can an equation y+p(t)y+q(t)y=0, with continuous...Ch. 4.2 - If the Wronskian W of f and g is 3e2t, and if...Ch. 4.2 - If the Wronskian W of f and g is t2et, and if...Ch. 4.2 - If W[f,g] is the Wronskian of f and g, and if...Ch. 4.2 - If the Wronskian of f and g is tcostsint, and if...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - In each of problem 22 through 25, verify that the...Ch. 4.2 - 26. Consider the equation (a). Show that and ...Ch. 4.2 - 27. Prove Theorem 4.2.4 and Corollary 4.2.5....Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - In each of problem 28 through 38, use method of...Ch. 4.2 - 37. The differential equation Where N is...Ch. 4.2 - The differential equation y+(xy+y)=0 arises in the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 1 through 26: (a) Find the...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - In each of Problems 27 through 43, solve the given...Ch. 4.3 - In each of Problems through, solve the given...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - Find a differential equation whose general...Ch. 4.3 - In each of Problems and , determine the values of...Ch. 4.3 - In each of Problems 47 and 48, determine the...Ch. 4.3 - If the roots of the characteristic equation are...Ch. 4.3 - Consider the equation ay+by+cy=d, where a,b,c and...Ch. 4.3 - Consider the equation , where and are constants...Ch. 4.3 - Prob. 52PCh. 4.3 - If , use the substitution to show that the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 54 through 61, find the...Ch. 4.3 - In each of Problems through, find the general...Ch. 4.3 - In each of Problems 62 through 65, find the...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.3 - In each of Problems through, find the solution of...Ch. 4.4 - In each of Problems through , determine and so...Ch. 4.4 - In each of Problems through , determine and so...Ch. 4.4 - In each of Problems 1 through 4, determine 0,R,...Ch. 4.4 - In each of Problems 1 through 4, determine 0,R,...Ch. 4.4 - (a) A mass weighing lb stretches a spring in. If...Ch. 4.4 - (a) A mass of 100 g stretches a spring 5 cm. If...Ch. 4.4 - A mass weighing 3 lb stretches a spring 3 in. If...Ch. 4.4 - A series circuit has a capacitor of 0.25...Ch. 4.4 - (a) A mass of g stretches a spring cm. Suppose...Ch. 4.4 - A mass weighing 16 lb stretches a spring 3in. The...Ch. 4.4 - (a) A spring is stretched cm by a force of ...Ch. 4.4 - (a) A series circuit has a capacitor of farad, a...Ch. 4.4 - A certain vibrating system satisfies the equation...Ch. 4.4 - Show that the period of motion of an undamped...Ch. 4.4 - Show that the solution of the initial value...Ch. 4.4 - Show that Acos0t+Bsin0t can be written in the form...Ch. 4.4 - A mass weighing 8 lb stretches a spring 1.5 in....Ch. 4.4 - If a series circuit has a capacitor of C=0.8...Ch. 4.4 - Assume that the system described by the equation...Ch. 4.4 - Assume that the system described by the equation...Ch. 4.4 - Logarithmic Decrement For the damped oscillation...Ch. 4.4 - Referring to Problem , find the logarithmic...Ch. 4.4 - For the system in Problem , suppose that and ....Ch. 4.4 - The position of a certain spring-mass system...Ch. 4.4 - Consider the initial value problem . We wish to...Ch. 4.4 - Consider the initial value problem...Ch. 4.4 - Use the differential equation derived in Problem...Ch. 4.4 - Draw the phase portrait for the dynamical system...Ch. 4.4 - The position of a certain undamped spring-mass...Ch. 4.4 - The position of a certain spring-mass system...Ch. 4.4 - In the absence of damping, the motion of a...Ch. 4.4 - If the restoring force of a nonlinear spring...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 1 through 16, find the general...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 17 through 22, find the...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - In each of problems 23 through 30: Determine a...Ch. 4.5 - Consider the equation (i) From...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Nonhomogeneous Cauchy-Euler Equations. In each of...Ch. 4.5 - Determine the general solution of , Where and ...Ch. 4.5 - In many physical problems, the nonhomogeneous term...Ch. 4.5 - Follow the instructions in Problem 37 to solve the...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - In each of Problems 1 through 4, write the given...Ch. 4.6 - A mass weighing 4 pounds (lb) stretches a spring...Ch. 4.6 - A mass of 4 kg stretches a spring 8 cm. The mass...Ch. 4.6 - (a) Find the solution of Problem 5. (b) Plot the...Ch. 4.6 - 8. Find the solution of the initial value problem...Ch. 4.6 - If an undamped spring-mass system with a mass that...Ch. 4.6 - A mass that weighs 8 lb stretches a spring 24 in....Ch. 4.6 - A spring is stretched 6 in. by a mass that weighs...Ch. 4.6 - A spring-mass system has a spring constant of 3...Ch. 4.6 - Furnish the details in determining when the gain...Ch. 4.6 - Find the solution of the initial value problem...Ch. 4.6 - A series circuit has a capacitor of 0.25...Ch. 4.6 - 16. Consider a vibrating system described by the...Ch. 4.6 - Consider the forced but undamped system described...Ch. 4.6 - Consider the vibrating system described by the...Ch. 4.6 - For the initial value problem in Problem 18, plot ...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - Problems 20 through 22 deal with the initial value...Ch. 4.6 - A spring-mass system with a hardening spring...Ch. 4.6 - Suppose that the system of Problem 23 is modified...Ch. 4.7 - (a) If and , show that . (b) Assuming that is...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 2 through 5, use the method of...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 6 through 9, find the solution...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 10 through 13, use the method...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 14 through 21, find the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 22 through 27, verify that the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - In each of Problems 28 through 31, find the...Ch. 4.7 - Show that the solution of the initial value...Ch. 4.7 - By choosing the lower limit of integration in Eq....Ch. 4.7 - (a) Use the result of Problem 33 to show that...Ch. 4.7 - Use the result of Problem 33 to find the solution...Ch. 4.7 - Use the result of Problem 33 to find the...Ch. 4.7 - Use the result of Problem 33 to find the solution...Ch. 4.7 - By combining the results of the problems 35...Ch. 4.7 - The method of reduction of order (see the...Ch. 4.7 - In each of problems 40 and 41, use the method...Ch. 4.7 - In each of problems and , use the method outlined...Ch. 4.P1 - Denote by the displacement of the platform from...Ch. 4.P1 - Denote by the frequency response of , that is,...Ch. 4.P1 - Plot the graphs of versus the dimensionless ratio...Ch. 4.P1 - The vibrations in the floor of an industrial plant...Ch. 4.P1 - Test the results of your design strategy for the...Ch. 4.P2 - Show that the differential equation describing the...Ch. 4.P2 - (a) Find the linearization of at . (b) In the...Ch. 4.P2 - Subject to the initial conditions , draw the graph...Ch. 4.P3 - Assuming that both springs have spring constant ...Ch. 4.P3 - The Heaviside, or unit step function, is defined...Ch. 4.P3 - Is the differential equation derived in Problems ...Ch. 4.P3 - In the case that the damping constant 0, find the...Ch. 4.P3 - Consider the case of an undamped problem using...Ch. 4.P3 - Consider the damped problem using the parameter...Ch. 4.P3 - Describe some other physical problems that could...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems 1 through 3 are concerned with one...Ch. 4.P4 - Problems and are concerned with systems that...Ch. 4.P4 - Problems and are concerned with systems that...Ch. 4.P4 - Carry out the calculations that lead from Eq. to...

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