Chapter 6: Q1E (page 332)

Find a general solution for the differential equation with x as the independent variable.
$y''' + 2 y'' - 8 y' = 0$

Short Answer

Expert verified

Thus, the general solution to the given differential equation is. $y = C_{1} + C_{2} e^{- 4 x} + C_{3} e^{2 x}$

Step by step solution

Use the given equation to find a general solution for the differential equation with x

The given differential equation is,

$y''' + 2 y'' - 8 y' = 0$

The auxiliary equation is. $m^{3} + 2 m^{2} - 8 m = 0$

Find the roots of the auxiliary equation.

$\begin{matrix} m^{3} + 2 m^{2} - 8 m = 0 \\ m (m^{2} + 2 m - 8) = 0 \\ m (m + 4) (m - 2) = 0 \\ m_{1} = 0, m_{2} = - 4, m_{3} = 2 \end{matrix}$

General solution.

The roots are real and distinct; therefore the general solution to the given differential equation is given as:

$\begin{matrix} y = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x} + C_{3} e^{m_{3} x} \\ y = C_{1} e^{(0) x} + C_{2} e^{(- 4) x} + C_{3} e^{(2) x} \\ y = C_{1} + C_{2} e^{- 4 x} + C_{3} e^{2 x} \end{matrix}$

Thus, the general solution to the given differential equation is. $y = C_{1} + C_{2} e^{- 4 x} + C_{3} e^{2 x}$

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Find a general solution for the differential equation with x as the independent variable.
$y''' + 2 y'' - 8 y' = 0$

Short Answer

Step by step solution

Use the given equation to find a general solution for the differential equation with x

General solution.

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Find a general solution for the differential equation with x as the independent variable.y'''+2y''-8y'=0

Short Answer

Step by step solution

Use the given equation to find a general solution for the differential equation with x

General solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Applied Mathematics

Probability and Statistics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.

Find a general solution for the differential equation with x as the independent variable.
$y''' + 2 y'' - 8 y' = 0$