Q20E In Problems 13鈥�20, solve the g... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q20E (page 164)

In Problems 13鈥�20, solve the given initial value problem.
y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1

Short Answer

Expert verified

The solution is y(t) = 2e^2t-2- te^3t+2.

Step by step solution

Find the solution of the differential equation.

The given differential equation is y" - 4y' + 4y = 0.

The auxiliary equation is r²- 4r + 4 = 0

Find the roots of the auxiliary equation;

$\begin{array}{rcl} r^{2} - 4 r + 4 & = & 0 \\ r & = & \frac{4 卤 \sqrt{{(- 4)}^{2} - 4 (1) (4)}}{2 (1)} \\ r & = & \frac{4 卤 \sqrt{16 - 16}}{2} \\ r & = & 2, 2 \end{array}$

Therefore the solution is y(t) = c₁e^2t + c₂te^2t.

Apply initial conditions.

The initial conditions are y(1) = 1, y'(t) = 1.

Therefore,

$\begin{array}{rcl} y (1) & = & c_{1} e^{2} + c_{2} (1) e^{2} \\ c_{1} e^{2} + c_{2} (1) e^{2} & = & 1 \\ c_{1} + c_{2} & = & e^{- 2} \end{array}$

And

$\begin{array}{rcl} y' (t) & = & 2 c_{1} e^{(2 t)} + c_{2} (2 t e^{(2 t)} + e^{2 t}) \\ y' (1) & = & 2 c_{1} e^{2} + c_{2} (2 (1) e^{2} + e^{2}) \\ 2 c_{1} e^{2} + c_{2} (3 e^{2}) & = & 1 \\ 2 c_{1} + 3 c_{2} & = & e^{- 2} \end{array}$

Solving for c₁,c₂ then;

c₁ = 2e^-2

c₂ = -e²

Therefore, the solution is $y (t) = 2 e^{- 2} e^{2 t} - e^{2} t e^{3 t} 鈬� y (t) = 2 e^{2 t - 2} - t e^{3 t + 2}$ .

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91影视

In Problems 13鈥�20, solve the given initial value problem.
y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1

Short Answer

Step by step solution

Find the solution of the differential equation.

Apply initial conditions.

One App. One Place for Learning.

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Recommended explanations on Math Textbooks

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91影视

In Problems 13鈥�20, solve the given initial value problem.y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1

Short Answer

Step by step solution

Find the solution of the differential equation.

Apply initial conditions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 13鈥�20, solve the given initial value problem.
y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1