Chapter 4: Q16RP (page 231)

Question: Find a general solution to the given differential equation. $y''' + 3 y'' + 5 y' + 3 y = 0$

Short Answer

Expert verified

The general solution to the given differential equation is:

$y = c_{1} e^{- t} + c_{2} e^{- t} \cos (\sqrt{2} t) + c_{3} e^{- t} \sin (\sqrt{2} t)$

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots, then the general solution is given as:

$y (t) = c_{1} e^{α t} c o s β t + c_{2} e^{α t} s i n β t$

Write the auxiliary equation of the given differential equation

The differential equation is $y''' + 3 y'' + 5 y' + 3 y = 0 .$

The auxiliary equation for the above equation $m^{3} + 3 m^{2} + 5 m + 3 = 0 .$

Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} m^{3} + 3 m^{2} + 5 m + 3 = 0 \\ m^{3} + m^{2} + 2 m^{2} + 2 m + 3 m + 3 = 0 \\ (m + 1) (m^{2} + 2 m + 3) = 0 \\ m = - 1, m = \frac{- 2 \pm \sqrt{4 - 12}}{2} \\ m = - 1, m = - 1 \pm i \sqrt{2} \end{matrix}$

The roots of the auxiliary equation are, $m_{1} = - 1, m_{2} = - 1 + i \sqrt{2}, m_{3} = - 1 - i \sqrt{2} .$

The general solution of the given equation is,

$y = c_{1} e^{- t} + c_{2} e^{- t} \cos (\sqrt{2} t) + c_{3} e^{- t} \sin (\sqrt{2} t)$

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Question: Find a general solution to the given differential equation. $y''' + 3 y'' + 5 y' + 3 y = 0$

Short Answer

Step by step solution

Complex conjugate roots.

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.y'''+3y''+5y'+3y=0

Short Answer

Step by step solution

Complex conjugate roots.

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

Mechanics Maths

Applied Mathematics

Calculus

Discrete Mathematics

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Question: Find a general solution to the given differential equation. $y''' + 3 y'' + 5 y' + 3 y = 0$