Chapter 4: Q20E (page 164)

In Problems 13–20, 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;

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

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 $y (t) = 2 e^{- 2} e^{2 t} - e^{2} t e^{3 t} \Rightarrow y (t) = 2 e^{2 t - 2} - t e^{3 t + 2}$ .

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In Problems 13–20, 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.

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Most popular questions from this chapter

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In Problems 13–20, 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

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Geometry

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

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