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

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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.
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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...
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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.

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

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