Explanation Given: The differential equation is, y ″ + y ′ − y = 3 t 4 − 3 t 3 + t . Calculation: Consider the second order differential equation. y ″ + y ′ − y = 3 t 4 − 3 t 3 + t (1) The characteristic polynomial is, r 2 + r − 1 = 0 . ( r − ( 1 + 5 2 ) ) ( r − ( 1 − 5 2 ) ) = 0 Therefore, the general solution is a combination of linear combination of exponential function. Here, the nonhomogeneous part of the equation is a polynomial highest degree four. The function 3 t 4 − 3 t 3 + t is not solution of the homogeneous part of the equation. Thus the trial solution is y = A t 4 + B t 3 + C t 2 + D t + E (2) Compute the value of A , B , C D and E , Differentiate y with respect to t . y ′ = d d t ( A t 4 + B t 3 + C t 2 + D t + E ) = 4 A t 3 + 3 B t 2 + 2 C t + D Differentiate y ′ with respect to t . y ″ = d d t ( 4 A t 3 + 3 B t 2 + 2 C t + D ) = 12 A t 2 + 6 B t + 2 C Substitute the values of y ″ , y ′ and y in equation (1). 3 t 4 − 3 t 3 + t = y ″ + y ′ − y = ( 12 A t 2 + 6 B t + 2 C ) + ( 4 A t 3 + 3 B t 2 + 2 C t + D ) − ( A t 4 + B t 3 + C t 2 + D t + E ) = 12 A t 2 + 6 B t + 2 C + 4 A t 3 + 3 B t 2 + 2 C t + D ȡ

Calculus: Early Transcendentals, 2nd Edition

2nd Edition

ISBN: 9780321965165

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter D2.3, Problem 43E

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To find: The particular solution of the given differential equation.

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Chapter D2.3, Problem 42E

Chapter D2.3, Problem 44E

Chapter D2 Solutions

Calculus: Early Transcendentals, 2nd Edition

Ch. D2.1 - Describe how to find the order of a differential...Ch. D2.1 - Prob. 2E Ch. D2.1 - Prob. 3E Ch. D2.1 - Give a general form of a second-order linear...Ch. D2.1 - Prob. 5E Ch. D2.1 - Prob. 6E Ch. D2.1 - Prob. 7E Ch. D2.1 - Prob. 8E Ch. D2.1 - Prob. 9E Ch. D2.1 - Prob. 10E

Ch. D2.1 - Prob. 11E Ch. D2.1 - Prob. 12E Ch. D2.1 - Prob. 13E Ch. D2.1 - Verifying solutions Verify by substitution that...Ch. D2.1 - Prob. 15E Ch. D2.1 - Prob. 16E Ch. D2.1 - Prob. 17E Ch. D2.1 - Prob. 18E Ch. D2.1 - Prob. 19E Ch. D2.1 - Prob. 20E Ch. D2.1 - Prob. 21E Ch. D2.1 - Prob. 22E Ch. D2.1 - Prob. 23E Ch. D2.1 - Prob. 24E Ch. D2.1 - Prob. 25E Ch. D2.1 - Prob. 26E Ch. D2.1 - Prob. 27E Ch. D2.1 - Prob. 28E Ch. D2.1 - Prob. 29E Ch. D2.1 - Prob. 30E Ch. D2.1 - Prob. 31E Ch. D2.1 - Prob. 32E Ch. D2.1 - Prob. 33E Ch. D2.1 - Prob. 34E Ch. D2.1 - Prob. 35E Ch. D2.1 - Prob. 36E Ch. D2.1 - Prob. 37E Ch. D2.1 - Prob. 38E Ch. D2.1 - Prob. 39E Ch. D2.1 - Prob. 40E Ch. D2.1 - Prob. 41E Ch. D2.1 - Prob. 42E Ch. D2.1 - Prob. 43E Ch. D2.1 - Initial value problems Solve the following initial...Ch. D2.1 - Prob. 45E Ch. D2.1 - Prob. 46E Ch. D2.1 - Explain why or why not Determine whether the...Ch. D2.1 - Prob. 48E Ch. D2.1 - Solution verification Verify by substitution that...Ch. D2.1 - Prob. 50E Ch. D2.1 - Prob. 51E Ch. D2.1 - Prob. 52E Ch. D2.1 - Prob. 53E Ch. D2.1 - Prob. 54E Ch. D2.1 - Prob. 55E Ch. D2.1 - Prob. 56E Ch. D2.1 - Prob. 57E Ch. D2.1 - Prob. 58E Ch. D2.1 - Prob. 59E Ch. D2.1 - Prob. 60E Ch. D2.1 - Prob. 61E Ch. D2.1 - Prob. 62E Ch. D2.1 - Prob. 63E Ch. D2.1 - Prob. 64E Ch. D2.1 - Prob. 65E Ch. D2.1 - Prob. 66E Ch. D2.1 - Prob. 67E Ch. D2.1 - Prob. 68E Ch. D2.1 - Prob. 69E Ch. D2.1 - Reduction of order Suppose you are solving a...Ch. D2.2 - Prob. 1E Ch. D2.2 - Prob. 2E Ch. D2.2 - Prob. 3E Ch. D2.2 - Prob. 4E Ch. D2.2 - Prob. 5E Ch. D2.2 - Prob. 6E Ch. D2.2 - Prob. 7E Ch. D2.2 - Give the trial solution used to solve a...Ch. D2.2 - Prob. 9E Ch. D2.2 - Prob. 10E Ch. D2.2 - General solutions with distinct real roots Find...Ch. D2.2 - Prob. 12E Ch. D2.2 - Prob. 13E Ch. D2.2 - Prob. 14E Ch. D2.2 - Initial value problems with distinct real roots...Ch. D2.2 - Prob. 16E Ch. D2.2 - Prob. 17E Ch. D2.2 - Prob. 18E Ch. D2.2 - Prob. 19E Ch. D2.2 - Prob. 20E Ch. D2.2 - Prob. 21E Ch. D2.2 - Prob. 22E Ch. D2.2 - Prob. 23E Ch. D2.2 - Prob. 24E Ch. D2.2 - Prob. 25E Ch. D2.2 - Prob. 26E Ch. D2.2 - Prob. 27E Ch. D2.2 - Prob. 28E Ch. D2.2 - Prob. 29E Ch. D2.2 - Prob. 30E Ch. D2.2 - Prob. 31E Ch. D2.2 - Prob. 32E Ch. D2.2 - Prob. 33E Ch. D2.2 - Prob. 34E Ch. D2.2 - Initial value problems with Cauchy-Euler equations...Ch. D2.2 - Prob. 36E Ch. D2.2 - Prob. 37E Ch. D2.2 - Initial value problems with Cauchy-Euler equations...Ch. D2.2 - Prob. 39E Ch. D2.2 - Prob. 42E Ch. D2.2 - Prob. 43E Ch. D2.2 - Prob. 44E Ch. D2.2 - Prob. 45E Ch. D2.2 - Prob. 46E Ch. D2.2 - Prob. 47E Ch. D2.2 - Prob. 48E Ch. D2.2 - Prob. 49E Ch. D2.2 - Prob. 50E Ch. D2.2 - Prob. 51E Ch. D2.2 - Cauchy-Euler equation with repeated roots It can...Ch. D2.2 - Prob. 53E Ch. D2.2 - Prob. 54E Ch. D2.2 - Prob. 55E Ch. D2.2 - Prob. 56E Ch. D2.2 - Prob. 57E Ch. D2.2 - Prob. 58E Ch. D2.2 - Prob. 59E Ch. D2.2 - Prob. 60E Ch. D2.2 - Prob. 61E Ch. D2.2 - Cauchy-Euler equation with repeated roots One of...Ch. D2.2 - Prob. 63E Ch. D2.2 - Prob. 64E Ch. D2.2 - Prob. 65E Ch. D2.2 - Prob. 66E Ch. D2.3 - Explain how to find the general solution of the...Ch. D2.3 - Prob. 2E Ch. D2.3 - Prob. 3E Ch. D2.3 - Prob. 4E Ch. D2.3 - Prob. 5E Ch. D2.3 - Prob. 6E Ch. D2.3 - Prob. 7E Ch. D2.3 - Prob. 8E Ch. D2.3 - Prob. 9E Ch. D2.3 - Prob. 10E Ch. D2.3 - Prob. 11E Ch. D2.3 - Prob. 12E Ch. D2.3 - Prob. 13E Ch. D2.3 - Undetermined coefficients with exponentials Find a...Ch. D2.3 - Prob. 15E Ch. D2.3 - Prob. 16E Ch. D2.3 - Prob. 17E Ch. D2.3 - Prob. 18E Ch. D2.3 - Prob. 19E Ch. D2.3 - Prob. 20E Ch. D2.3 - Prob. 21E Ch. D2.3 - Prob. 22E Ch. D2.3 - Prob. 23E Ch. D2.3 - Prob. 24E Ch. D2.3 - Prob. 25E Ch. D2.3 - Prob. 26E Ch. D2.3 - Prob. 27E Ch. D2.3 - Prob. 28E Ch. D2.3 - Prob. 29E Ch. D2.3 - Prob. 30E Ch. D2.3 - Prob. 31E Ch. D2.3 - Prob. 32E Ch. D2.3 - Prob. 33E Ch. D2.3 - Prob. 34E Ch. D2.3 - Prob. 35E Ch. D2.3 - Prob. 36E Ch. D2.3 - Prob. 37E Ch. D2.3 - Initial value problems Find the general solution...Ch. D2.3 - Prob. 39E Ch. D2.3 - Prob. 40E Ch. D2.3 - Prob. 41E Ch. D2.3 - Prob. 42E Ch. D2.3 - Prob. 43E Ch. D2.3 - Prob. 44E Ch. D2.3 - Prob. 45E Ch. D2.3 - Prob. 46E Ch. D2.3 - Prob. 47E Ch. D2.3 - Prob. 48E Ch. D2.3 - Prob. 49E Ch. D2.3 - Prob. 50E Ch. D2.3 - Prob. 51E Ch. D2.3 - Variation of parameters Finding a particular...Ch. D2.4 - Explain the meaning of the words damped, undamped,...Ch. D2.4 - In the models discussed in this section, under...Ch. D2.4 - Prob. 3E Ch. D2.4 - Prob. 4E Ch. D2.4 - Prob. 5E Ch. D2.4 - Prob. 6E Ch. D2.4 - Prob. 7E Ch. D2.4 - Prob. 8E Ch. D2.4 - Prob. 9E Ch. D2.4 - Free undamped oscillations Solve the initial value...Ch. D2.4 - Prob. 11E Ch. D2.4 - Prob. 12E Ch. D2.4 - Prob. 13E Ch. D2.4 - Prob. 14E Ch. D2.4 - Prob. 15E Ch. D2.4 - Prob. 16E Ch. D2.4 - Free damped oscillations Solve the initial value...Ch. D2.4 - Free damped oscillations Solve the initial value...Ch. D2.4 - Designing a shock absorber A shock absorber must...Ch. D2.4 - Designing a suspension system A spring in a...Ch. D2.4 - Forced damped oscillations 21.A 1-kg block hangs...Ch. D2.4 - Forced damped oscillations 22.A 20-kg block hangs...Ch. D2.4 - Prob. 23E Ch. D2.4 - Prob. 24E Ch. D2.4 - Prob. 25E Ch. D2.4 - Prob. 26E Ch. D2.4 - Prob. 27E Ch. D2.4 - LCR circuits 28.The circuit in Exercise 27 (10-ohm...Ch. D2.4 - Prob. 29E Ch. D2.4 - Prob. 30E Ch. D2.4 - Prob. 31E Ch. D2.4 - LCR circuits 32.Find the charge on the capacitor...Ch. D2.4 - Explain why or why not Determine whether the...Ch. D2.4 - Prob. 34E Ch. D2.4 - Prob. 35E Ch. D2.4 - Prob. 36E Ch. D2.4 - Prob. 37E Ch. D2.4 - Prob. 38E Ch. D2.4 - Prob. 39E Ch. D2.4 - Prob. 41E Ch. D2.4 - Prob. 42E Ch. D2.4 - Prob. 43E Ch. D2.4 - Prob. 44E Ch. D2.4 - Applications 4346.Horizontal oscillators The...Ch. D2.4 - Prob. 46E Ch. D2.4 - Prob. 47E Ch. D2.4 - Prob. 48E Ch. D2.4 - Prob. 49E Ch. D2.4 - Prob. 51E Ch. D2.4 - Prob. 52E Ch. D2.5 - Prob. 1E Ch. D2.5 - Prob. 2E Ch. D2.5 - Prob. 3E Ch. D2.5 - Prob. 4E Ch. D2.5 - Prob. 5E Ch. D2.5 - Prob. 6E Ch. D2.5 - Prob. 7E Ch. D2.5 - Prob. 8E Ch. D2.5 - Gain and phase lag functions Consider the...Ch. D2.5 - Prob. 10E Ch. D2.5 - Prob. 11E Ch. D2.5 - Solutions to oscillator equations Consider the...Ch. D2.5 - Prob. 13E Ch. D2.5 - Solutions to oscillator equations Consider the...Ch. D2.5 - Prob. 15E Ch. D2.5 - Prob. 16E Ch. D2.5 - Prob. 17E Ch. D2.5 - Prob. 18E Ch. D2.5 - Analyzing circuit equations Consider the circuit...Ch. D2.5 - Prob. 20E Ch. D2.5 - Prob. 21E Ch. D2.5 - Prob. 22E Ch. D2.5 - Prob. 23E Ch. D2.5 - A high-pass filter Consider the LCR circuit shown...Ch. D2.5 - High-pass filters Consider the high-pass filter...Ch. D2.5 - Prob. 26E Ch. D2.5 - High-pass filters Consider the high-pass filter...Ch. D2.5 - Prob. 28E Ch. D2 - Prob. 1RE Ch. D2 - Prob. 2RE Ch. D2 - Prob. 3RE Ch. D2 - Prob. 4RE Ch. D2 - Solving homogeneous equations Find the general...Ch. D2 - Prob. 6RE Ch. D2 - Prob. 7RE Ch. D2 - Prob. 8RE Ch. D2 - Prob. 9RE Ch. D2 - Prob. 10RE Ch. D2 - Prob. 11RE Ch. D2 - Prob. 12RE Ch. D2 - Prob. 13RE Ch. D2 - Prob. 14RE Ch. D2 - Prob. 15RE Ch. D2 - Prob. 16RE Ch. D2 - Prob. 17RE Ch. D2 - Prob. 18RE Ch. D2 - Prob. 19RE Ch. D2 - Prob. 20RE Ch. D2 - Prob. 21RE Ch. D2 - Forced undamped oscillations A 4-kg block hangs on...Ch. D2 - Free damped oscillations A 0.2-kg block hangs on a...

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The particular solution of the given differential equation.

Solution Summary: The author explains that the general solution of the second order differential equation is a combination of linear and exponential functions.

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To find: The particular solution of the given differential equation.

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Chapter D2 Solutions