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Chapter 8: Q5.2P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$y " - 4 y' + 4 y = 0$

Short Answer

Expert verified

The solution is $y = (A x + B) e^{2 x}$

Step by step solution

01

Given information

A differential equation is given as $y'' - 4 y' + 4 y = 0$ .

02

Differential equation.

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

$y = (A x + B) e^{a x}$

03

Find the solution of given differential equation.

A differential equation is given as $y " - 4 y' + 4 y = 0$ .

Calculation:

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 $y " - 4 y' + 4 y = 0$ .

Above equation becomes,

role="math" localid="1664376798686" $D^{2} y - 4 D y + 4 y = 0 (D^{2} - 4 D + 4) y = 0 (D - 2) (D - 2) y = 0$

$(D - 2) y = 0$ and $(D - 2) y = 0$

The general solution will be given by $y = (A x + B) e^{2 x}$

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Most popular questions from this chapter

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See all solutions

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