Chapter 4: Q38E (page 165)

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it. $y''' - 6 y'' - y' + 6 y = 0$ .

Short Answer

Expert verified

$y (t) = c_{1} e^{t} + c_{2} e^{- t} + c_{3} e^{6 t}$ is the solution of the given equation $y''' - 6 y'' - y' + 6 y = 0$ .

Step by step solution

Differentiate the value of y

Given differential equation is $y''' - 6 y'' - y' + 6 y = 0$

Let $y = e^{r t}$ , then $y' (t) = r e^{r t}$

$y'' (t) = r^{2} e^{r t} y''' = r^{3} e^{r t}$

Then the auxiliary equation is:

$r^{3} e^{r t} - 6 r^{2} e^{r t} - r e^{r t} + 6 e^{r t} = 0 (r^{3} - 6 r^{2} - r + 6) e^{r t} = 0 r^{3} - 6 r^{2} - r + 6 = 0$

Substitute the value for r

Let substitute $r = 1$

$1 - 6 - 1 + 6 = 0$

Therefore $r = 1$ is the solution of the auxiliary equation $r^{3} - 6 r^{2} - r + 6 = (r - 1) (r^{2} - 5 r - 6)$

$(r - 1) (r^{2} - 5 r - 6) = 0$

Hence, the solutions are $r = 1$ and $(r + 1) (r - 6) = 0$

$r = 1, - 1, 6 y (t) = c_{1} e^{t} + c_{2} e^{- t} + c_{3} e^{6 t}$

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Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it. $y''' - 6 y'' - y' + 6 y = 0$ .

Short Answer

Step by step solution

Differentiate the value of y

Substitute the value for r

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Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.y'''-6y''-y'+6y=0.

Short Answer

Step by step solution

Differentiate the value of y

Substitute the value for r

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it. $y''' - 6 y'' - y' + 6 y = 0$ .