Chapter 6: Q7E (page 332)

Find a general solution for the differential equation with x as the independent variable:
$2 y''' - y'' - 10 y' - 7 y = 0$

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variableis . $y = C_{1} e^{- x} + C_{2} e^{\frac{3 + \sqrt{65}}{4} x} + C_{3} e^{\frac{3 - \sqrt{65}}{4} x}$

Step by step solution

Auxiliary equation:

The given differential equationis $2 y''' - y'' - 10 y' - 7 y = 0$ . To solve this equation, we look at its auxiliary equation which is $2 m^{3} - m^{2} - 10 m - 7 = 0$ . Observe that -1 is a solution of this equation. So, $2 m^{3} - m^{2} - 10 m - 7 = (m + 1) (2 m^{2} - 3 m - 7)$

Inspecting the sum further:

To get the other two roots of auxillary equation, we need to solve $2 m^{2} - 3 m - 7 = (m + 1) (2 m^{2} - 3 m - 7)$ . We have,

$m = \frac{3 \pm \sqrt{9 + 56}}{4} = \frac{3 \pm \sqrt{65}}{4}$

General solution:

We have m = -1, $\frac{3 \pm \sqrt{65}}{4}$ . From (7) of 328 and (18) of page 330, we conclude that the general solution of the given differential equation is $y = C_{1} e^{- x} + C_{2} e^{\frac{3 + \sqrt{65}}{4} x} + C_{3} e^{\frac{3 - \sqrt{65}}{4} x}$ where $C_{1}, C_{2}, C_{3}$ are arbitrary constants.

The solution of the given differential equation is $y = C_{1} e^{- x} + C_{2} e^{\frac{3 + \sqrt{65}}{4} x} + C_{3} e^{\frac{3 - \sqrt{65}}{4} x}$ , where $C_{1}, C_{2}, C_{3}$ are arbitrary constant.

Hence the final solution is $y = C_{1} e^{- x} + C_{2} e^{\frac{3 + \sqrt{65}}{4} x} + C_{3} e^{\frac{3 - \sqrt{65}}{4} x}$

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Find a general solution for the differential equation with x as the independent variable:
$2 y''' - y'' - 10 y' - 7 y = 0$

Short Answer

Step by step solution

Auxiliary equation:

Inspecting the sum further:

General solution:

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Find a general solution for the differential equation with x as the independent variable:2y'''−y''−10y'−7y=0

Short Answer

Step by step solution

Auxiliary equation:

Inspecting the sum further:

General solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find a general solution for the differential equation with x as the independent variable:
$2 y''' - y'' - 10 y' - 7 y = 0$