Explanation Given: The differential equation is, y ‴ + y ″ − 2 y = t e t ... ( 1 ) Approach: To find a particular solution to the differential equation a y ″ + b y ′ + c y = C t m e r t where m is a nonnegative integer, use the form y p ( t ) = t 5 ( A m t m + ..... + A 1 t 1 + A 0 ) e r t with s = 0 if r is not a root of the associated auxiliary equation; s = 1 if r is simple root of the associated auxiliary equation and, s = 2 if r is double root of the associated auxiliary equation. Calculation: The auxiliary equation of equation ( 1 ) is as follows. r 3 + r 2 − 2 = 0 . Solve the auxiliary equation for roots. r 3 − r 2 + 2 = 0 r 3 − 2 r 2 + r 2 + 2 r − 2 r + 2 = 0 ( r + 1 ) ( r 2 − 2 r + 2 ) = 0 So, the roots are. r = − 1 , 2 ± 4 − 8 2 = − 1 , 2 ± 2 i 2 = − 1 , 1 ± i Here, one root is real and two roots are imaginary. Suppose the roots of the above auxiliary equation are, α = 1 , β = 1 and r = − 1 . The complementary solution involves the term e t due to the root α = 1 . By undetermined coefficients, the particular solution is y p ( t ) = t ( A t + B ) e t y p ( t ) = ( A t 2 + B t ) e t ... ( 2 ) Differentiate the above equation. y ′ p ( t ) = d d t ( A t 2 + B t ) e t y ′ p ( t ) = ( A t 2 + B t ) e t + e t ( 2 A t + B ) y ′ p ( t ) = e t ( A t 2 + ( 2 A + B ) t + B ) Again differentiate the above equation. y ″ p ( t ) = d d t e t ( A t 2 + ( 2 A + B ) t + B ) y ″ p ( t ) = e t ( A t 2 + ( 2 A + B ) t + B ) + e t ( 2 A t + ( 2 A + B ) ) y ″ p ( t ) = e t [ ( A t 2 + ( 2 A + B ) t + B ) + 2 A t + ( 2 A + B ) ] y ″ p ( t ) = e t ( A t 2 + ( 4 A + B ) t + 2 A + 2 B ) Again differentiate the above equation

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

See similar textbooks

Concept explainers

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…

Videos

Question

Chapter 4.4, Problem 35E

To determine

To find:

The particular solution to the differential equation.

y‴+y″−2y=tet

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 34E

Chapter 4.4, Problem 36E

Students have asked these similar questions

The joint pdf of random variables X=1, 2 and Y=1, 2, 3 is P(X,Y)= X 10.05 Find (a) The value of k. (c) P(X>1, Y <2). Y 0.2 0.18 0.15] (b) the marginal probability function of X and Y. (d) Ex, Hy

The conditional probability function for the random variables X and Y is 0 P(Y/X) = x0 [0.9 10.1 y 1 2 0.1 0 0.8 0.1 2 0 0.1 0.9. With P(x=0)=0.2, P(x-1)=0.4. Find P(X,Y), Hx, My, E(XY), OXY.

If X is a continuous random variable having pdf as shown. Find 1. The constant k. 2. P(X>0). 3. X, X2,0%. k p(x) 4 k/2 X -3 -1 0 1 2

Chapter 4 Solutions

EBK FUND.OF DIFF.EQUATIONS+BOUNDARY...

Ch. 4.1 - Verify that for b=0 and Fext(t)=0, equation (3)...Ch. 4.1 - If Fext(t)=0, equation (3) becomes my+by+ky=0. For...Ch. 4.1 - Show that if Fext(t)=0, m=1, k=9, and b=6, then...Ch. 4.1 - Prob. 4E Ch. 4.1 - Verify that the exponentially damped sinusoid...Ch. 4.1 - An external force F(t)=2cos2t is applied to a...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Undamped oscillators that are driven at resonance...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.