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Chapter 8: Q5.8P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$D (D + 5) y = 0$

Short Answer

Expert verified

The solution is $y = c_{1} + c_{2} e^{- 5 x}$

Step by step solution

01

Given information

A differential equation is given as $D (D + 5) y = 0$ .

02

Differential equation.

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

$y = c_{1} e^{a x} + c_{2} e^{b x} .$

03

Find the solution of the 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 $D (D + 5) y = 0$ .

Auxiliary equation is-

$D (D + 5) = 0$

These are unequal roots $鈬� D = 0, D = - 5$

Differential equation becomes

$鈬� (D - 0) [D - (- 5)] y = 0$

These are separable equation with solution

$y = c_{1} e^{0 x} and y = c_{2} e^{- 5 x}$

The general solution will be given by

$y = c_{1} e^{0 x} + c_{2} e^{- 5 x} y = c_{1} + c_{2} e^{- 5 x}$

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Most popular questions from this chapter

Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are $N_{0}$ bacteria when $t = 0$ . Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.

Find the shape of a mirror which has the property that rays from a point 0 on the axis are reflected into a parallel beam. Hint: Take the point 0 at the origin. Show from the figure that $t a n 胃 = 2 \frac{y}{x}$ . Use the formula for $t a n 胃 2$ to express this in terms of $t a n 胃 = \frac{d y}{d x}$ and solve the resulting differential equation. (Hint: See Problem 16.)

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

$y y' - 2 y^{2} c o t x = s i n x c o s x$

Using Problems 29 and 31b, show that equation (6.24) is correct.

Several Terms on the Right-Hand Side: Principle of Superposition So far we have brushed over a question which may have occurred to you: What do we do if there are several terms on the right-hand side of the equation involving different exponentials?

In Problem 33 to 38 , solve the given differential equations by using the principle of superposition [see the solution of equation (6.29) . For example, in Problem 33 , solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus a polynomial of any degree is kept together in one bracket.

$y^{"} - 5 y^{'} + 6 y = 2 e^{x} + 6 x - 5$

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

$\overset{篓}{y} + 4 \overset{藱}{y} + 5 y = 2 e^{- 2 t} c o s t, y_{0} = 0, {\overset{藱}{y}}_{0} = 3$

See all solutions

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