please help with all three, section 1.6= Second-order constant-coefficient homogeneousequations section 1.7 = Second-order constant-coefficient nonhomo-geneous equations, one photo is the question the others are examples 1.29 and 1.3 Example 1.29. Let a and b be constants, and fi(t) and f2(t) be functions oft. Show that if yp, (t) is a particular solution to y" +ay'+by = f(t) and yp₂ (t)is a particular solution to y" + ay'+by = f(t), then a particular solution toy" + ay' + by = f(t) + f2(t) is given by yp(t) = yp₁ (t) + yp2 (t).We can do this by simple verification. Let us formy+ay+byp==[+] + a[yp₁ + yp₂] + b[yp₁ + yp₂][yp₁ + ayp₁ + byp₁] + [yº₂ + ay₂ + byp2]fi(t) + f2(t),and thus the function y(t) does indeed satisfy the given equation.Example 1.30. Using the above two examples as a guide, what would be theappropriate form for the undetermined coefficients solution for the equationy"(t)- y' (t) - 6y(t) = -7e³t+etcos (2t) ? (a) Use the methods from section 1.6 to find the characteristic equa-tion and the general solution to the homogeneous differentialequationd²ydr²+ 4y = 0.(b) Use the methods from section 1.7 to write down the correct guessfor the form you would use to find a particular solution to thenonhomogeneous differential equationd'ydr²[Hint: Examples 1.29 and 1.30 might be helpful here.]+ 4y = cos(4x) + cos(2x).

According to bartleby guifdelines we supposed to do the first one of multiple questions kindly…

Answered: (a) Use the methods from section 1.6 to…

(a) Use the methods from section 1.6 to find the characteristic equa- tion and the general solution to the homogeneous differential equation d'y dr2 + 4y = 0.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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