(a) To determine To show: The exact solution of the initial value problem y ′ = 1 − t + y , y ( t 0 ) = y 0 is y = ϕ ( t ) = ( y 0 − t 0 ) e t − t 0 + t . Explanation The given initial value problem is y ′ = 1 − t + y , y ( t 0 ) = y 0 . Rewrite the differential equation as shown below. d y d t = 1 − t + y d y d t − y = 1 − t Compare the above linear equation with d y d t + P ( t ) y = G ( t ) and obtain P ( t ) = − 1 , G ( t ) = 1 − t . The general solution of d y d t + P ( t ) y = G ( t ) is e ∫ P ( t ) d t y = ∫ G ( t ) e ∫ P ( t ) d t d t + C where e ∫ P ( t ) d y is the integrating factor. Compute the integrating factor for d y d t − y = 1 − t as follows. e ∫ P ( t ) d t = e ∫ ( − 1 ) d t = e − 1 ∫ d t = e − t Now the solution of y ′ = 1 − t + y is obtained as shown below. e − t y = ∫ ( 1 − t ) e − t d t + C = ∫ e − t d t − ∫ t e − t d t + C = ∫ e − t d t − ∫ t d ( e − t − 1 ) + C = − e − t + t (b) To determine To show: The equation y k = ( 1 + h ) y k − 1 + h − h t k − 1 , k = 1 , 2 , … by using Euler formula. (c) To determine To show: The equation y n = ( 1 + h ) n ( y 0 − t 0 ) + t n by method of induction provided y 1 = ( 1 + h ) ( y 0 − t 0 ) + t 1 . (d) To determine To show: The value of y n → ϕ ( t ) as n → ∞ , by choosing h = t − t 0 n .

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.7, Problem 20P

(a)

To determine

To show: The exact solution of the initial value problem $y′=1−t+y, y(t0)=y0$ is $y=ϕ(t)=(y0−t0)et−t0+t$ .

(b)

To determine

To show: The equation $yk=(1+h)yk−1+h−htk−1, k=1,2,…$ by using Euler formula.

(c)

To determine

To show: The equation $yn=(1+h)n(y0−t0)+tn$ by method of induction provided $y1=(1+h)(y0−t0)+t1$ .

(d)

To determine

To show: The value of $yn→ϕ(t) as n→∞$ , by choosing $h=t−t0n$ .

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Chapter 2.7, Problem 19P

Chapter 2.7, Problem 21P

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