Chapter 6: Q17E (page 332)

Find a general solution to the givenhomogeneous equation. $(D + 4) (D - 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$

Short Answer

Expert verified

The answer to this problem is:

$y = c_{1} e^{- 4 x} + c_{2} e^{3 x} + (c_{3} + c_{4} x + c_{5} x^{2}) e^{- 2 x} + (c_{6} + c_{7} x) e^{- 2 x} \cos x + (c_{8} + c_{9} x) \sin x + c_{10} + c_{11} x + c_{12} x^{2} + c_{13} x^{3} + c_{14} x^{4}$

Step by step solution

Homogenous Equation

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous.

Solving of Homogenous Equation:

We will have to solve the following homogenous equation here;

$\begin{array}{l} D + 4) (D - 3) {(D + 2)}^{2} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0 \\ (r + 4) (r - 3) {(r + 2)}^{3} {(r^{2} + 4 r + 5)}^{2} r^{5} = 0 \\ r_{1} = - 4 \\ r_{2} = 3 \\ r_{3, 4, 5} = - 2 \\ r^{2} + 4 r + 5 = 0 \\ r_{6, 7, 8, 9} = \frac{- 4 \pm \sqrt{4^{2} - 4.5}}{2} \\ = - 2 \pm i \\ r_{10, 11, 12, 13, 14} = 0 \\ y = c_{1} e^{- 4 x} + c_{2} e^{3 x} + (c_{3} + c_{4} x + c_{5} x^{2}) e^{- 2 x} + (c_{6} + c_{7} x) e^{- 2 x} \cos x + (c_{8} + c_{9} x) \sin x + c_{10} + c_{11} x + c_{12} x^{2} + c_{13} x^{3} + c_{14} x^{4} \end{array}$

Hence, the final answer is:

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Find a general solution to the givenhomogeneous equation. $(D + 4) (D - 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$

Short Answer

Step by step solution

Homogenous Equation

Solving of Homogenous Equation:

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Find a general solution to the givenhomogeneous equation.(D+4)(D−3)(D+2)3(D2+4D+5)2D5[y]=0

Short Answer

Step by step solution

Homogenous Equation

Solving of Homogenous Equation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Find a general solution to the givenhomogeneous equation. $(D + 4) (D - 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$