{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

nagle_differential_equations_ISM_Part30

# nagle_differential_equations_ISM_Part30 - Exercises 4.6 So...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Exercises 4.6 So, y p ( t ) =- 1 2 t 2 ln t- 1 4 t 2 e- 2 t + t (ln t- 1) Â· te- 2 t = 2 ln t- 3 4 t 2 e- 2 t , and a general solution is given by y ( t ) = 2 ln t- 3 4 t 2 e- 2 t + c 1 e- 2 t + c 2 te- 2 t . 12. The corresponding homogeneous equation is y 00 + y = 0. Its auxiliary equation has the roots r = Â± i . Hence, a general solution to the homogeneous corresponding problem is given by y h = c 1 cos t + c 2 sin t. We will find a particular solution to the original equation by representing the right-hand side as a sum tan t + e 3 t- 1 = g 1 ( t ) + g 2 ( t ) , where g 1 ( t ) = tan t and g 2 ( t ) = e 3 t- 1. A particular solution to y 00 + y = g 1 ( t ) was found in Example 1, namely, y p, 1 =- (cos t ) ln | sec t + tan t | . A particular solution to y 00 + y = g 2 ( t ) can be found using the method of undetermined coefficients. We let y p, 2 = A e 3 t + B â‡’ y 00 p, 2 = 9 A e 3 t . Substituting these functions yields y 00 p, 2 + y p, 2 = ( 9 A e 3 t ) + ( A e 3 t + B ) = 10 A e 3 t + B = e 3 t- 1 . Hence, A = 1 / 10, B =- 1, and y p, 2 = (1 / 10) e 3 t- 1. By the superposition principle, y = y p, 1 + y p, 2 + y h =- (cos t ) ln | sec t + tan t | + (1 / 10) e 3 t- 1 + c 1 cos t + c 2 sin t gives a general solution to the original equation. 141 Chapter 4 14. A fundamental solution set for the corresponding homogeneous equation is y 1 ( Î¸ ) = cos Î¸ and y 2 ( Î¸ ) = sin Î¸ (see Example 1 in the text or Problem 12). Applying the method of variation of parameters, we seek a particular solution to the given equation in the form y p = v 1 y 1 + v 2 y 2 , where...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

nagle_differential_equations_ISM_Part30 - Exercises 4.6 So...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online