Chapter 6: Q3E (page 332)

Find a general solution for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$

Short Answer

Expert verified

Thus, the general solution to the given differential equation is;

z = C_{1} e^{- x} + C_{2} e^{\frac{x}{2}} + C_{3} e^{\frac{- 2 x}{3}}

Step by step solution

Write the auxiliary equation.

The given differential equation is;

$6 z''' + 7 z'' - z' - 2 z = 0$

The auxiliary equation is. $6 m^{3} + 7 m^{2} - m - 2 = 0$

Simplify the auxiliary equation.

$\begin{matrix} 6 m^{3} + 7 m^{2} - m - 2 = 0 \\ 6 m^{3} + 7 m^{2} - m - 2 = 0 \\ (m + 1) (6 m^{2} + m - 2) = 0 \end{matrix}$

One of the roots is . $m_{1} = - 1$

Find the other roots of the auxiliary equation by solving the quadratic equation.

$\begin{matrix} m = \frac{- 1 \pm \sqrt{1 + 4 (12)}}{12} \\ m = \frac{- 1 \pm \sqrt{49}}{12} \\ m = \frac{- 1 \pm 7}{12} \\ m = \frac{- 1 + 7}{12}, \frac{- 1 - 7}{12} \\ m_{2} = \frac{1}{2}, m_{3} = - \frac{2}{3} \end{matrix}$

Write the general solution.

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

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

Thus, the general solution to the given differential equation is;

$z = C_{1} e^{- x} + C_{2} e^{\frac{x}{2}} + C_{3} e^{\frac{- 2 x}{3}}$

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Find a general solution for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$

Short Answer

Step by step solution

Write the auxiliary equation.

Write the general solution.

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Find a general solution for the differential equation with x as the independent variable.6z'''+7z''-z'-2z=0

Short Answer

Step by step solution

Write the auxiliary equation.

Write the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Statistics

Decision Maths

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Find a general solution for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$