Chapter 4: Q12RP (page 231)

Question: Find a general solution to the given differential equation. $2 y''' - 3 y'' - 12 y' + 20 y = 0$

Short Answer

Expert verified

$y = c_{1} e^{2 t} + c_{2} {te}^{2 t} + c_{3} e^{\frac{- 5}{2} t}$

Step by step solution

Write the auxiliary equation of the given differential equation

The differential equation is $2 y''' - 3 y'' - 12 y' + 20 y = 0 .$

The auxiliary equation for the above equation $2 m^{3} - 3 m^{2} - 12 m + 20 = 0 .$

Now find the roots of the auxiliary equation.

Solve the auxiliary equation,

$\begin{matrix} 2 m^{3} - 3 m^{2} - 12 m + 20 = 0 \\ 2 m^{3} + 5 m^{2} - 8 m^{2} - 20 m + 8 m + 20 = 0 \\ m^{2} (2 m + 5) - 4 m (2 m + 5) + 4 (2 m + 5) = 0 \\ (2 m + 5) (m^{2} - 4 m + 4) = 0 \\ (2 m + 5) {(m - 2)}^{2} = 0 \end{matrix}$

The roots of the auxiliary equation are, $m_{1} = 2, m_{2} = 2, m_{3} = - \frac{5}{2} .$

Thus, the general solution of the given equation is $y = (c_{1} + c_{2} t) e^{2 t} + c_{3} e^{\frac{- 5}{2} t} .$

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Question: Find a general solution to the given differential equation. $2 y''' - 3 y'' - 12 y' + 20 y = 0$

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation

Now find the roots of the auxiliary equation.

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Question: Find a general solution to the given differential equation.2y'''-3y''-12y'+20y=0

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation

Now find the roots of the auxiliary equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Question: Find a general solution to the given differential equation. $2 y''' - 3 y'' - 12 y' + 20 y = 0$