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Unformatted text preview: December 9, 2010 11 Exam 3 and later course material review These pages give a summary of the material required for Exam 3. Subject Matter: 1. Laplace transforms (a) L ( f ( t ) for p ( t )(polynomial) ,sin ( at ) ,cos ( at ) , e at f ( t ) ,t n f ( t ) ,sinh ( at ) ,cosh ( at ) ,f ( t ) ? g ( t ) , u c ( t ) f ( t ) (b) Inverse Laplace transforms: use of partial fractions and complet ing the square (c) Piecewise smooth functions (use of u c ( t )) (d) Initial value problems with Laplace transforms ay 00 + by + cy = h ( t ) , y (0) = y ,y (0) = y L ( y ( t )) = ( as + b ) y , + ay + L ( h ( t )) as 2 + bs + c Take inverse transform to get y ( t ). 2. Algebraic Systems of Linear Equations, row reduction method, Cramers rule, determinants, computation of inverses 3. Two dimensional Systems of linear differential equations. (a) Homogeneous equations: x = ax + by y = cx + dy x 1 = ax 1 + bx 2 x 2 = cx 1 + dx 2 x = a b c d ! x , x = x 1 x 2 ! December 9, 2010 12 A = a b c d ! characteristic polynomial z ( r ) = det ( rI A ) = r 2 ( a + d ) r + ad bc Three cases: i. Distinct real roots r 1 ,r 2 : General solution: x ( t ) = c 1 e r 1 t v 1 + c 2 e r 2 t v 2 , v 1 6 = 6 = v 2 Av 1 = r 1 v 1 , Av 2 = r 2 v 2 If b 6 = 0, then v 1 = 1 r 1 a b !...
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