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Chapter 8: Q5.10P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions.
$y^{''} - 2 y^{'} = 0$

Short Answer

Expert verified

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

Step by step solution

01

Given information

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

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 $y^{''} - 2 y^{'} = 0$ .

$D^{2} y - 2 D y = 0 (D^{2} - 2 D) y = 0$

Auxiliary equation is-

$D^{2} - 2 D = 0 D (D - 2) = 0 D = 0 D = 2$

These are unequal roots $鈬� D = 0, D = 2$

Differential equation becomes

$鈬� (D - 0) [D - 2] y = 0$

These are separable equation with solution

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

The general solution will be given by

$y = c_{1} e^{0 x} + c_{2} e^{2 x} 鈬� y = c_{1} + c_{2} e^{2 x}$

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Most popular questions from this chapter

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

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

Sketch on the same axes graphs of $s i n t, s i n (t - 蟺 / 2)$ , and $s i n (t + 蟺 / 2)$ , and observe which way the graph shifts. Hint: You can, of course, have your calculator or computer plot these for you, but it's simpler and much more useful to do it in your head. Hint: What values of $t$ make the sines equal to zero? For an even simpler example, sketch on the same axes $y = t, y = t - 蟺 / 2, y = t + 蟺 / 2$ .

Use L32 and L11 to obtain $L (t^{2} s i n a t)$ .

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

$y' = c o s (x + y)$

Hint: Let $u = x + y$ ; then $u' = 1 + y'$ .

Evaluate each of the following definite integrals by using the Laplace transform table.

See all solutions

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