Explanation Given: The differential equation is 10 y ″ + y ′ − 3 y = t − e t / 2 . Approach: If a n y ( n ) + a n − 1 y ( n − 1 ) + ⋅ ⋅ ⋅ + a 1 y ( 1 ) + a 0 = p ( t ) , then y = y c +y p . Here, y c denotes the complementary solution, y p denotes the particular solution. The complementary solution is the solution of the differential equation a n y ( n ) + a n − 1 y ( n − 1 ) + ⋅ ⋅ ⋅ + a 1 y ( 1 ) + a 0 = 0 . The particular solution is the possible linear combinations of the function p ( t ) . The auxiliary equation of a differential equation a n y ( n ) + a n − 1 y ( n − 1 ) + ⋅ ⋅ ⋅ + a 1 y ( 1 ) + a 0 is a n r n + a n − 1 r n − 1 + ⋅ ⋅ ⋅ + a 1 r 1 + a 0 ... ( 1 ) . Here, y ( n ) denotes the n th derivative of y . If x = a is a zero of the polynomial a n x n + a n − 1 x n − 1 + ⋅ ⋅ ⋅ + a 1 x 1 + a 0 , then ( x − a ) is a factor of the polynomial ... ( 2 ) . If the auxiliary equation has complex roots r 1 = a + b i , r 2 = a − b i then the solution of the differential equation is y = A e a t cos b t + B e a t sin b t ... ( 3 ) Calculation: The differential equation is y ″ + y ′ 10 − 3 10 = 1 10 ( t − e t 2 ) . Consider y ″ + y ′ 10 − 3 10 = 0 The auxiliary equation of the differential equation is r 2 + 1 10 r − 3 = 0 Let p ( r ) = r 2 + 1 10 r − 3 Put p ( r ) = 0 r 2 + 1 10 r − 3 = 0 10 r 2 + r − 3 = 0 ( 2 r − 1 ) ( 5 r + 3 ) = 0 The roots of p ( r ) are 1 2 , − 3 5 . By ( 2 ) , y h ( t ) = C 1 e 1 2 t + C 2 e − 3 5 t . p ( t ) = t − e t / 2 . The particular solution is the linear combination of t − e t / 2 and its possible derivatives. y p ( t ) = A + B t e t 2 + C t . Differentiate y p ( t ) 2 times with respect to t . y ′ p ( t ) = B ( t × 1 2 e t 2 + e t 2 × 1 ) + C y ′ p ( t ) = B e t 2 ( t 2 + 1 ) + C y ″ p ( t ) = B ( e t 2 × 1 2 + 1 2 × e t 2 × ( t 2 + 1 ) ) = B ( e t 2 + 1 4 t e t 2 ) Since, y p is a solution of y ″ − 3 y ′ + 7 y = 7 t 2 − e t By ( 2 ) , y ″ p − 3 y ′ p + 7 y p = 7 t 2 − e t Substitute A + B t e t 2 + C t for y p , B e t 2 ( t 2 + 1 ) + C for y ′ p , B ( e t 2 + 1 4 t e t 2 ) for y ″ p in y ″ + y ′ 10 − 3 10 = 1 10 ( t − e t 2 )

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.RP, Problem 24RP

To determine

To find:

A general solution to the differential equation

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.RP, Problem 23RP

Chapter 4.RP, Problem 25RP

Students have asked these similar questions

7. [10] Suppose that Xi, i = 1,..., 5, are independent normal random variables, where X1, X2 and X3 have the same distribution N(1, 2) and X4 and X5 have the same distribution N(-1, 1). Let (a) Find V(X5 - X3). 1 = √(x1 + x2) — — (Xx3 + x4 + X5). (b) Find the distribution of Y. (c) Find Cov(X2 - X1, Y). -

1. [10] Suppose that X ~N(-2, 4). Let Y = 3X-1. (a) Find the distribution of Y. Show your work. (b) Find P(-8< Y < 15) by using the CDF, (2), of the standard normal distribu- tion. (c) Find the 0.05th right-tail percentage point (i.e., the 0.95th quantile) of the distri- bution of Y.

6. [10] Let X, Y and Z be random variables. Suppose that E(X) = E(Y) = 1, E(Z) = 2, V(X) = 1, V(Y) = V(Z) = 4, Cov(X,Y) = -1, Cov(X, Z) = 0.5, and Cov(Y, Z) = -2. 2 (a) Find V(XY+2Z). (b) Find Cov(-x+2Y+Z, -Y-2Z).

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.

A general solution to the differential equation

Solution Summary: The author explains how to find a general solution to the differential equation: y(t)=C_1e

Concept explainers

Videos

To find: A general solution to the differential equation

Want to see the full answer?

Chapter 4 Solutions

To find:

A general solution to the differential equation