Q47 E In quantum mechanics, the study ... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q47 E (page 201)

In quantum mechanics, the study of the Schr枚dinger equation for the case of a harmonic oscillator leads to a consideration of Hermite's equation, $y'' - 2 ty' + 位测 = 0$ , where $位$ is a parameter. Use the reduction of order formula to obtain an integral representation of a second linearly independent solution to Hermite's equation for the given value of $位$ and the corresponding solution $f (t)$ .
$(a) 位 = 4, f (t) = 1 - 2 t^{2} (b) 位 = 6, f (t) = 3 t - 2 t^{3}$

Short Answer

Expert verified

The second linearly independent solution of the given Hermite鈥檚 equation $位 = 4, f (t) = 1 - 2 t^{2}$ is $y = c_{1} (1 - 2 t^{2}) + c_{2} (1 - 2 t^{2}) 鈭� \frac{e^{t^{2}}}{{(1 - 2 t^{2})}^{2}} dt$ .
The second linearly independent solution of the given Hermite鈥檚 equation $位 = 6, f (t) = 3 t - 2 t^{3}$ is $y = c_{1} (3 t - 2 t^{3}) + c_{2} (3 t - 2 t^{3}) 鈭� \frac{e^{t^{2}}}{{(3 t - 2 t^{3})}^{2}} dt$ .

Step by step solution

Finding the value of Y

Given differential equation is $y'' - 2 ty' + 位测 = 0$ and here $位 = 4$ and $f (t) = 1 - 2 t^{2}$ is one of the solutions and $p (t) = - 2 t$ . Then

$y_{2} = y_{1} (t) 鈭� \frac{e^{- 鈭� p (t)} dt}{y_{1}^{2} (t)} dt = (1 - 2 t^{2}) 鈭� \frac{e^{- 鈭� - 2 t} dt}{{(1 - 2 t^{2})}^{2}} dt = (1 - 2 t^{2}) 鈭� \frac{e^{t^{2}}}{{(1 - 2 t^{2})}^{2}} dt$

So, the solution is $y = c_{1} (1 - 2 t^{2}) + c_{2} (1 - 2 t^{2}) 鈭� \frac{e^{t^{2}}}{{(1 - 2 t^{2})}^{2}} dt$ .

Finding the value of Y

Here $位 = 6$ and $f (t) = 1 - 2 t^{2}$ is one of the solutions and $p (t) = 3 t - 2 t^{3}$ . Then

$y_{2} = y_{1} (t) 鈭� \frac{e^{- 鈭� p (t)} dt}{y_{1}^{2} (t)} dt = (3 t - 2 t^{3}) 鈭� \frac{e^{- 鈭� - 2 t} dt}{{(3 t - 2 t^{3})}^{2}} dt = (3 t - 2 t^{3}) 鈭� \frac{e^{t^{2}}}{{(3 t - 2 t^{3})}^{2}} dt$