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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: Q46 E (page 201)

Find a particular solution to the non-homogeneous equation $(1 - t) y'' + ty' - y = (1 - t)^{2}$ , given that $f (t) = t$ is a solution to the corresponding homogeneous equation.

Short Answer

Expert verified

The solution to the given non-homogeneous equation $(1 - t) y'' + ty' - y = (1 - t)^{2}$ is $y = c_{1} e^{t} + c_{2} t$

.

Step by step solution

01

Find the value of Y

Given differential equation is $(1 - t) y'' + ty' - y = (1 - t)^{2}$ then the standard form of the solution is $y'' + \frac{t}{1 - t} y' - \frac{1}{1 - t} y = \frac{1}{1 - t}$ and $f (t) = t$ is one of the solutions and $p (t) = \frac{t}{1 - t}$

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

02

Simplification

Simplify the above.

$t 鈭� \frac{e^{- 鈭� \frac{t}{1 - t}} dt}{{(t)}^{2}} dt = t 鈭� \frac{e^{\ln (t - 1) + t}}{t^{2}} = t 脳鈭� \frac{(t - 1) 脳 e^{t}}{t^{2}} = t 脳 \frac{e^{t}}{t} = e^{t}$

So, the solution is $y = c_{1} e^{t} + c_{2} t$

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