Chapter 4: Q16E (page 199)

Find a general solution for $t < 0$ .
$t^{2} y'' (t) - 3 ty' (t) + 6 y (t) = 0$

Short Answer

Expert verified

The general solution of the given equation $t^{2} y'' (t) - 3 ty' (t) + 6 y (t) = 0$ is:

$y = c_{1} t^{2} \cos (\sqrt{2} lnt) + c_{2} t^{2} \sin (\sqrt{2} lnt)$ for $t < 0$ .

Step by step solution

Substitute the values

Given differential equation is $t^{2} y'' (t) - 3 ty' (t) + 6 y (t) = 0$

Assume $y = t^{r}$ , then $y' = {rt}^{r - 1}$

$y'' = r (r - 1) t^{r - 2}$

Substitute these equations in the differential equation

$\begin{array}{rcl} t^{2} r (r - 1) t^{r - 2} - 3 {trt}^{r - 1} + 6 t^{r} & = & 0 \\ (r (r - 1) - 3 r + 6) t^{t} & = & 0 \\ r^{2} - 4 r + 6 & = & 0 \end{array}$

Finding roots

Find the roots of this equation

$r = \frac{4 \pm \sqrt{4^{2} - 4 \times 6 \times 1}}{2 \times 1} r = \frac{4 \pm \sqrt{16 - 24}}{2} r = \frac{4 \pm \sqrt{- 8}}{2} r = \frac{4 \pm 2 \sqrt{2} i}{2} r = 2 \pm \sqrt{2} i$

Therefore, the general solution is $y = c_{1} t^{2} \cos (\sqrt{2} lnt) + c_{2} t^{2} \sin (\sqrt{2} lnt)$ .

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.

Find a general solution for $t < 0$ .
$t^{2} y'' (t) - 3 ty' (t) + 6 y (t) = 0$

Short Answer

Step by step solution

Substitute the values

Finding roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Mechanics Maths

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find a general solution for t<0.t2y''(t)-3ty'(t)+6y(t)=0

Short Answer

Step by step solution

Substitute the values

Finding roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Mechanics Maths

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find a general solution for $t < 0$ .
$t^{2} y'' (t) - 3 ty' (t) + 6 y (t) = 0$