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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 6: Q25E (page 337)

. use the annihilator method to determinethe form of a particular solution for the given equation.
$y'' - 6 y' + 9 y = s i n 2 x + x$

Short Answer

Expert verified

$y_{p} (x) = c_{1} + c_{2} x + c_{5} \cos 2 x + c_{6} \sin 2 x$

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

$(D^{2} - 6 D + 9) [y] = 0$

The solution of the homogeneous is

$y_{h} (x) = c_{1} e^{3 x} + c_{2} x e^{3 x}$ (1)

Now $\sin 2 x + x$ is annihilated by $(D^{4} + 4 D^{2})$

Then, every solution to the given nonhomogeneous equation also satisfies

. $(D^{4} + 4 D^{2}) (D^{2} - 6 D + 9) [y] = 0$

Then

$y (x) = c_{1} + c_{2} x + c_{3} e^{3 x} + c_{4} x e^{3 x} +$ $c_{5} \cos 2 x + c_{6} \sin 2 x$ (2)

is the general solution to this homogeneous equation

We know $u (x) = u_{h} + u_{p}$

Comparing (1) & (2)

$y_{p} (x) = c_{1} + c_{2} x + c_{5} \cos 2 x + c_{6} \sin 2 x$

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Most popular questions from this chapter

Determine the intervals for which Theorem guarantees the existence of a solution in that

(a) $y^{(4)} - (l n x) y^{''} + x y^{'} + 2 y = c o s 3 x$

(b) $(x^{2} - 1) y^{'''} + (s i n x) y^{''} + \sqrt{x + 4} y^{'} + e^{x} y = x^{2} + 3$

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${\sin^{2} x, 鈥� \cos^{2} x, 鈥� 1}$ on $(- 鈭�, 鈥� 鈭�)$

find a general solution to the given equation.

$y''' - 3 y'' + 3 y' - y = e^{x}$

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}}$ on $(- 鈭�, 鈥� 鈭�)$

Use the annihilator method to show that if $f (x)$ in (4) has the form $f (x) = a c o s 尾 x + b s i n 尾 x,$

then equation (4) has a particular solution of the form

(18) $y_{p} (x) = x^{s} {A c o s 尾 x + B s i n 尾 x}$ ,where sis chosen to be the smallest nonnegative integer such that $x^{3} c o s 尾 x$ and $x^{3} \sin 尾 x$ are not solutions to the corresponding homogeneous equation

See all solutions

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