Chapter 4: Q10E (page 164)

Question: find a general solution to the given differential equation $y'' - y' - 11 y = 0$ .

Short Answer

Expert verified

Answer

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

Step by step solution

Write the auxiliary equation of the given differential equation.

The given differential equation is $y'' - y' - 11 y = 0 .$

The auxiliary equation for the above equation $m^{2} - m - 11 = 0 .$

Now find the roots of the auxiliary equation.

Solve the auxiliary equation,

$m^{2} - m - 11 = 0 m = \frac{- (- 1) \pm \sqrt{1 - 4 (1) (- 11)}}{2} m = \frac{1 \pm \sqrt{45}}{2} m = \frac{1 \pm 3 \sqrt{5}}{2}$

The roots of the auxiliary equation are

$m_{1} = \frac{1 + 3 \sqrt{5}}{2}, & m_{2} = \frac{1 - 3 \sqrt{5}}{2} .$

Write the general solution.

If an auxiliary equation has distinct real roots&, then the general solution is given as;

$y = c_{1} e^{m_{1} t} + c_{2} e^{m_{2} t}$

Thus, the general solution of the given equation is

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Question: find a general solution to the given differential equation $y'' - y' - 11 y = 0$ .

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Write the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Probability and Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: find a general solution to the given differential equation y''-y'-11y=0.

Short Answer

Step by step solution

Write the auxiliary equation of the given differential equation.

Now find the roots of the auxiliary equation.

Write the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Probability and Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: find a general solution to the given differential equation $y'' - y' - 11 y = 0$ .