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(1) First verify that yn and 2 are solutions of the following differential equations, then find a particular solution of the form y=ch+2 that satisfies the given initial conditions. Primes denote derivatives with respect to z. (a) y-y=0; -, /2; y(0)-0, y'(0)-5. (b) z²" +2ry-6y=0; -², 2-³; (2)-10, (2) - 15.

(1) First verify that yn and 2 are solutions of the following differential equations, then find a particular solution of the form y=ch+2 that satisfies the given initial conditions. Primes denote derivatives with respect to z. (a) y-y=0; -, /2; y(0)-0, y'(0)-5. (b) z²" +2ry-6y=0; -², 2-³; (2)-10, (2) - 15.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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(1) First verify that y₁ and 2 are solutions of the following differential equations, then find a particular solution of the form y = ₁3/1+022 that satisfies the given initial conditions. Primes denote derivatives with respect to z. (a) y - y = 0; h=², 1/₂=²; y(0) = 0, y/'(0) = 5. (b) x²y" + 2xy-6y=0; ₁=², 1/₂ = ³; (2) = 10, (2) = 15. (2) Show that 1/1 = ² and 2 = ³ are two solutions of zy" - Ary' +6y= 0, both satisfying the initial conditions y(0) = 0 ='(0). Explain why these facts do not contradict Theorem 2 (with respect to the guaranteed uniqueness). (3) Find the general solutions of the differential equation: Ay" - 12y +9y=0. (4) Solve the initial value problem: (5) Solve the initial value problem: "-6y +25y=0; y(0) = 3, y(0) = 1. "+"=0; y(0) = -1, (0) = 0, "(0) = 1.
Transcribed Image Text:(1) First verify that y₁ and 2 are solutions of the following differential equations, then find a particular solution of the form y = ₁3/1+022 that satisfies the given initial conditions. Primes denote derivatives with respect to z. (a) y - y = 0; h=², 1/₂=²; y(0) = 0, y/'(0) = 5. (b) x²y" + 2xy-6y=0; ₁=², 1/₂ = ³; (2) = 10, (2) = 15. (2) Show that 1/1 = ² and 2 = ³ are two solutions of zy" - Ary' +6y= 0, both satisfying the initial conditions y(0) = 0 ='(0). Explain why these facts do not contradict Theorem 2 (with respect to the guaranteed uniqueness). (3) Find the general solutions of the differential equation: Ay" - 12y +9y=0. (4) Solve the initial value problem: (5) Solve the initial value problem: "-6y +25y=0; y(0) = 3, y(0) = 1. "+"=0; y(0) = -1, (0) = 0, "(0) = 1.
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