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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

01

Given information

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

02

Differential equation.

A differential equation of the form $(D - a) (D - b) y = 0, a, b = 伪卤尾 i$ has general solution

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

03

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

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

These are unequal roots $鈬� D = - 1 卤 i$ .

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

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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Most popular questions from this chapter

Prove the general formula L29.

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.

$y' c o s x + y = c o s^{2} x$

Verify the statement of Example 2. Also verify that $y = c o s h x$ and $y = s i n h x$ are solutions of $y^{''} = y$ .

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.

$y' + y t a n h x = 2 e^{x}$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " + 4 y' + 4 y = e - 2 t, y v = 0, 尉_{0} = 4$

See all solutions

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