Fundamentals of Calculus notes

Fundamentals of Calculus 1/ Chapters 1-217/ All chapters

Fundamentals of Calculus 1/ Chapters 1-217/ All Chapters

1. Find the solution to the following initial value problems. (a) y ′′′ − y ′′ − 4y ′ + 4y = 0; y(0) = −4, y′ (0) = −1, y′′(0) = −19. (b) y ′′′ − 4y ′′ + 7y ′ − 6y = 0; y(0) = 1, y′ (0) = 0, y′′(0) = 0. Solution (a) The characteristic polynomial for this d.e. is: C(r) = r 3 − r 2 − 4r + 4 = (r + 2)(r − 1)(r − 2), whose roots are r = −2, 1, 2. The general solution to the equation is: y(x) = c1e −2x + c2e x + c3e 2x ...

Fundamentals of Calculus 1/ Chapters 1-217/ All Chapters

Fundamentals of Calculus 1/ Chapters 1-217/ All Chapters

Fundamentals of Calculus 1/ Chapters 1-217/ All Chapters

Fundamentals of Calculus 1/ Chapters 1-217/ All chapters

1. Find the solution to the following initial value problems. (a) y ′′′ − y ′′ − 4y ′ + 4y = 0; y(0) = −4, y′ (0) = −1, y′′(0) = −19. (b) y ′′′ − 4y ′′ + 7y ′ − 6y = 0; y(0) = 1, y′ (0) = 0, y′′(0) = 0. Solution (a) The characteristic polynomial for this d.e. is: C(r) = r 3 − r 2 − 4r + 4 = (r + 2)(r − 1)(r − 2), whose roots are r = −2, 1, 2. The general solution to the equation is: y(x) = c1e −2x + c2e x + c3e 2x ...