Question

(a) Convert the third order equation $\frac{d^3 u}{d t^3}+3 \frac{d^2 u}{d t^2}+4 \frac{d u}{d t}+12 u=0$ into a first order system in three variables by setting $u_1=u, u_2=\dot{u}, u_3=\ddot{u}$. (b) Solve the equation directly, and then use this to write down the general solution to your first order system. (c) What is the dimension of the solution space?

    (a) Convert the third order equation $\frac{d^3 u}{d t^3}+3 \frac{d^2 u}{d t^2}+4 \frac{d u}{d t}+12 u=0$ into a first order system in three variables by setting $u_1=u, u_2=\dot{u}, u_3=\ddot{u}$. (b) Solve the equation directly, and then use this to write down the general solution to your first order system. (c) What is the dimension of the solution space?

Show more…

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 10, Problem 2

Instant Answer

Step 1

Given the third order differential equation: \[ \frac{d^3 u}{d t^3} + 3 \frac{d^2 u}{d t^2} + 4 \frac{d u}{d t} + 12 u = 0 \] Define new variables: \[ u_1 = u, \quad u_2 = \frac{du}{dt}, \quad u_3 = \frac{d^2 u}{dt^2} \] Then, express the derivatives of these new Show more…

Show all steps

lock

Thanks for your feedback!

(a) Convert the third order equation $\frac{d^3 u}{d t^3}+3 \frac{d^2 u}{d t^2}+4 \frac{d u}{d t}+12 u=0$ into a first order system in three variables by setting $u_1=u, u_2=\dot{u}, u_3=\ddot{u}$. (b) Solve the equation directly, and then use this to write down the general solution to your first order system. (c) What is the dimension of the solution space?

Play audio

Feedback

Powered by NumerAI

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later