Chapter 4: Q7RP (page 231)

Find a general solution to the given differential equation. $36 y'' + 24 y' + 5 y = 0$

Short Answer

Expert verified

The general solution to the given differential equation is:

$y = c_{1} e^{\frac{- 1}{3} t} \cos (\frac{t}{6}) + c_{2} e^{\frac{- 1}{3} t} \sin (\frac{t}{6})$

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 $36 y'' + 24 y' + 5 y = 0 .$

The auxiliary equation for the above equation $36 m^{2} + 24 m + 5 = 0 .$

Find the roots of the auxiliary equation.

Solve the auxiliary equation,

$\begin{matrix} 36 m^{2} + 24 m + 5 = 0 \\ m = \frac{- 24 \pm \sqrt{576 - 720}}{2 (36)} \\ m = \frac{- 24 \pm \sqrt{- 144}}{72} \\ m = \frac{- 24 \pm 12 i}{72} \\ m = \frac{- 1}{3} \pm \frac{i}{6} \end{matrix}$

The roots of the auxiliary equation are, $m_{1} = \frac{- 1}{3} + \frac{i}{6}, & m_{2} = \frac{- 1}{3} - \frac{i}{6} .$

Thus, the general solution of the given equation is $y = c_{1} e^{\frac{- 1}{3} t} \cos (\frac{t}{6}) + c_{2} e^{\frac{- 1}{3} t} \sin (\frac{t}{6}) .$

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Find a general solution to the given differential equation. $36 y'' + 24 y' + 5 y = 0$

Short Answer

Step by step solution

Complex conjugate roots.

Write the auxiliary equation of the given differential equation

Find the roots of the auxiliary equation.

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Find a general solution to the given differential equation.36y''+24y'+5y=0

Short Answer

Step by step solution

Complex conjugate roots.

Write the auxiliary equation of the given differential equation

Find the roots of the auxiliary equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Logic and Functions

Probability and Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find a general solution to the given differential equation. $36 y'' + 24 y' + 5 y = 0$