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Chapter 8: Q5.11P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$4 y^{''} + 12 y^{'} + 9 = 0$

Short Answer

Expert verified

The solution is $y = (A x + B) e^{- \frac{3}{2} x}$ .

Step by step solution

01

Given information

A differential equation is given as $4 y " + y' + 9 = 0$ .

02

Differential equation.

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

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

03

Find the solution of given differential equation.

A differential equation is given as $4 y "' + y' + 9 = 0$ .

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 $4 y " + y' + 9 = 0$ .

Above equation becomes,

$4 D^{2} y + 12 D y + 9 = 0 (2 D + 3) (2 D + 3) y = 0 (2 D + 3) y = 0 and (2 D + 3) y = 0 (D + \frac{3}{2}) y = 0 and (D + \frac{3}{2}) y = 0$

Solve the problem further and get

$鈬� [D - (- \frac{3}{2})] y = 0 and [D - (- \frac{3}{2})] y = 0$

The general solution will be given by $y = (A x + B) e^{- \frac{3}{2} x}$ .

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Most popular questions from this chapter

Verify that $y = s i n x$ ,role="math" localid="1654838724304" $y = c o s x$ role="math" localid="1654838779452" $y = e^{i x}$ , and $y = e^{鈭� i x}$ are all solutions of $y^{''} = 鈭� y$ .

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

13. Problem 2

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$y' = \frac{y}{x} - t a n \frac{y}{x}$

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

9 $(1 + y) y' = y,$ $y = 1$ When $x = 1$

Find the family of orthogonal trajectories of the circles ${(x - h)}^{2} + y^{2} = h^{2}$ . (See the instructions above Problem 2.31.)

See all solutions

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