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Chapter 8: Q7P (page 422)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$(D^{2} 鈭� 5 D + 6) y = 0$

Short Answer

Expert verified

The solution is. $y = c_{1} e^{2 x} + c_{2} e^{3 x}$

Step by step solution

01

Given information

A differential equation is given as. $(D^{2} 鈭� 5 D + 6) y = 0$

02

Differential equation.

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

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

03

Find the solution of given differential equation.

Supposethat. $D = \frac{d}{d x}$

Then

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

Given differential equation is. $(D^{2} 鈭� 5 D + 6) y = 0$

Auxiliary equation is-

$\begin{matrix} D^{2} 鈭� 5 D + 6 = 0 \\ (D 鈭� 2) (D 鈭� 3) = 0 \\ D = 2 \\ D = 3 \end{matrix}$

These are unequal roots.

$鈬� D = 2, D = 3$

Differential equation becomes

$鈬� (D 鈭� 2) (D 鈭� 3) y = 0$

These are separable equation with solution

$y = c_{1} e^{2 x} and y = c_{2} e^{3 x}$

The general solution will be givenby. $y = c_{1} e^{2 x} + c_{2} e^{3 x}$

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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.

$y d y + (x y^{2} - 8 x) d x = 0, y = 3$ when $x = 1$ .

Solve $(12.3)$ if $G = 0$ and $d G / d t = 0$ at $t = 0$ to obtain (12.5). Hint: Use L28 and L3 to find the inverse transform.

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

$y " + y = s i n t, 3 t = 0, y'_{0} = - \frac{1}{2}$ ,

Problems 2 and 3, use (12.6) to solve (12.1) when $f (t)$ is as given. $f (t) = s i n 蝇 t$

See all solutions

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