In this problem, y = c1ex + c2e-x is a two-parameter family of solutions of the second-order DE y'' - y = 0. Find c1 and c2 given the following initial conditions. (Your answers will not contain a variable.) y (1) = 0, y' (1) = e c1 = Incorrect: Your answer is incorrect. c2 = Incorrect: Your answer is incorrect. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. y = Incorrect: Your answer is incorrect.

Question

Augustus Nichols

Mathematics

25 September, 07:22

In this problem, y = c1ex + c2e-x is a two-parameter family of solutions of the second-order DE y'' - y = 0. Find c1 and c2 given the following initial conditions. (Your answers will not contain a variable.) y (1) = 0, y' (1) = e c1 = Incorrect: Your answer is incorrect. c2 = Incorrect: Your answer is incorrect. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. y = Incorrect: Your answer is incorrect.

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Answers (1)

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Jennifer Arias · Answer 1

c₁ = 1/2

c₂ = - e²/2

y = (1/2) * (eˣ - e²⁻ˣ)

Step-by-step explanation:

Given

y = c₁eˣ + c₂e⁻ˣ

y (1) = 0

y' (1) = e

We get y':

y' = (c₁eˣ + c₂e⁻ˣ) ' ⇒ y' = c₁eˣ - c₂e⁻ˣ

then we find y (1):

y (1) = c₁e¹ + c₂e⁻¹ = 0

⇒ c₁ = - c₂/e² (I)

then we obtain y' (1):

y' (1) = c₁e¹ - c₂e⁻¹ = e (II)

⇒ ( - c₂/e²) * e - c₂e⁻¹ = e

⇒ - c₂e⁻¹ - c₂e⁻¹ = - 2c₂e⁻¹ = e

⇒ c₂ = - e²/2

and

c₁ = - c₂/e² = - ( - e²/2) / e²

⇒ c₁ = 1/2

Finally, the equation will be

y = (1/2) * eˣ - (e²/2) * e⁻ˣ = (1/2) * (eˣ - e²⁻ˣ)