Chapter 8: Q5.4P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$y^{''} + 2 y^{'} + 2 y = 0$

Short Answer

Expert verified

The solution is $y = c_{1} e^{(- 1 + i) x} + c_{2} e^{(- 1 - i) x}$ .

Step by step solution

Given information

A differential equation is given as $y " + 2 y' + 2 y = 0$ .

Differential equation.

A differential equation of the form $(D - a) (D - b) y = 0, a, b = α \pm β i$ has general solution

$y = c_{1} e^{(α + β i) x} + c_{2} e^{(α - β i) x}$

Calculate the solution of given differential equation.

Suppose that $D = \frac{d}{d x}$ .

Then

$D y = \frac{d y}{d x} D y = y^{'} D^{2} y = \frac{d}{d x} (\frac{d y}{d x}) \frac{d^{2} y}{d x^{2}} = y^{''}$

Given differential equation is $y^{''} + 2 y^{'} + 2 y = 0$ .

The above equation becomes,

$D^{2} y + 2 D y + 2 y = 0 (D^{2} + 2 D + 2) y = 0$

The auxiliary equation is

$D^{2} + 2 D + 2 = 0 D = \frac{- 2 \pm \sqrt{2^{2} - 4 \cdot 1 \cdot 2}}{2 \cdot 1} D = \frac{- 2 \pm 2 i}{2} D = - 1 \pm i$

These are unequal roots $\Rightarrow D = - 1 \pm i$ .

The differential equation becomes $\Rightarrow [D - (- 1 + i)] [D - (- 1 - i)] y = 0$ -

These are two separable equations with solution $y = c_{1} e^{(- 1 + i) x}$ and $y = c_{2} e^{(- 1 - i) x}$

The general solution will be given by $y = c_{1} e^{(- 1 + i) x} + c_{2} e^{(- 1 - i) x}$ .

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Solve the following differential equations by the methods discussed above and compare computer solutions.
$y^{''} + 2 y^{'} + 2 y = 0$

Short Answer

Step by step solution

Given information

Differential equation.

Calculate the solution of given differential equation.

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Solve the following differential equations by the methods discussed above and compare computer solutions.y''+2y'+2y=0

Short Answer

Step by step solution

Given information

Differential equation.

Calculate the solution of given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Wave Optics

Medical Physics

Translational Dynamics

Electric Charge Field and Potential

Radiation

Study anywhere. Anytime. Across all devices.

Solve the following differential equations by the methods discussed above and compare computer solutions.
$y^{''} + 2 y^{'} + 2 y = 0$