10Ma2aPrHW3

# 10Ma2aPrHW3 - 1 4. If an equation P ( t ) y 00 + Q ( t ) y...

This preview shows pages 1–2. Sign up to view the full content.

Ma 2a P: Homework N.3 due Tuesday Oct 26, 12 noon 1. Determine if the following functions are linearly independent: f 1 ( t ) = cos(2 t ) - 2 cos 2 ( t ) and f 2 ( t ) = cos(2 t ) + 2 sin 2 ( t ); f 1 ( t ) = e 5 t and f 2 ( t ) = e 5( t +1) . 2. Let y 1 ( t ) and y 2 ( t ) be two solutions of the homogeneous second order equation y 00 + p ( t ) y 0 + q ( t ) y = 0 where p ( t ) and q ( t ) are continuous on an interval t I = ( α,β ). If the Wronskian of the two solutions is constant, what can one say about p ( t ) and q ( t )? Show that if y 1 ( t ) and y 2 ( t ) vanish at the same point in the interval I , or if they have a maximum or a minimum at the same point, then they are not the fundamental set of solutions. 3. A second order linear homogeneous equation P ( t ) y 00 + Q ( t ) y 0 + R ( t ) y = 0 is exact if it can be written in the form ( P ( t ) y 0 ) 0 + ( f ( t ) y ) 0 = 0 , for a function f ( t ) that depends on P ( t ), Q ( t ), and R ( t ). Write the solutions of an exact equations in terms of P ( t ), f ( t ) and integrals. Show that the condition P 00 - Q 0 + R = 0 is necessary for exactness.

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: 1 4. If an equation P ( t ) y 00 + Q ( t ) y + R ( t ) y = 0 is not exact, one can look for an integrating factor μ ( t ) such that μ ( t ) P ( t ) y 00 + μ ( t ) Q ( t ) y + μ ( t ) R ( t ) y = 0 is exact. • Show that such a factor μ ( t ) should satisfy the adjoint equation Pμ 00 + (2 P-Q ) μ + ( P 00-Q + R ) μ = 0 to ensure that the equation μ ( t ) P ( t ) y 00 + μ ( t ) Q ( t ) y + μ ( t ) R ( t ) y = 0 is exact. • Show that the adjoint equation of the adjoint equation is the orig-inal equation Py 00 + Qy + Ry = 0. • Compute the adjoint equation of the Airy equation y 00-ty = 0 and of the Bessel equation t 2 y 00 + ty + ( t 2-ν 2 ) y = 0. 2...
View Full Document

## This note was uploaded on 11/19/2010 for the course ANTH 122 taught by Professor 323 during the Spring '10 term at Centennial College.

### Page1 / 2

10Ma2aPrHW3 - 1 4. If an equation P ( t ) y 00 + Q ( t ) y...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online