FinalReview - AMATH 351 Final Review Note: this review only...

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Unformatted text preview: AMATH 351 Final Review Note: this review only summarizes key knowledge but not covers all the materials we learned. Chapter 5 Series Solutions Method Power Series ∑ Check convergence of a power series by ratio test, | | { ( )( Taylor expansion ( ) ∑ ) ( ) Use ratio test to decide radius of convergence Comparison of series solutions between O.P. and R.S.P. Consider differential equation () O.P () () () R.S.P () ) () ( () Form of series solution Recurrence Relation Indicial Equation ∑ ( ) ( ∑ ( () ( () ) ) and ρ Involves N/A Involves ) Practice Problems Check if ( ( Section 5.1 #7, 12 Section 5.2 #8 is a O.P. or R.S.P. for the following D.E.’s ( ) ) ) ( ) Section 5.6 #10 Chapter 6 Laplace Transform Definition For a function ( ), its Laplace transform is () if the above integral exists. Denoted as () () * ( )+ * ( )+ ∫ () Properties * () *( )+ ( )+ * ( )+ * ( )+ Also * () * ( )+ ( )+ * ( )+ * ( )+ * ( )+ * ( )+ { () } { () ( )} () () ( ) () ( ) () Particularly * ( )+ * ( )+ () () () () () Use Laplace Transform to solve D.E. Consider for example () Take the Laplace transform , Solve for ( ) ( )() () () () () () () ( ), () ( )() () () () () () () ( () () ) () Then making use of the table on page #, find the inverse Laplace transform, namely ( ), () * ( )+ { () ( () () ) () } Practice Problems Section 6.2 #5, 20 Chapter 7 System of 1st order differential equations Conversion from nth order D.E. to system of n 1st order D.E.’s () () ( ) () () () Make substitution ( ) Then the D.E. is equivalent to ̂ Where ( )̂ ̂( ) ̂ ( ) ̂( ) ( ) () () ( () () () ( )) The corresponding homogeneous equation is ̂( ) ̂( ) Basic Matrix Operations Addition Subtraction scalar multiplication Matrix multiplication inverse matrix eigenvalue and eigenvector For a square matrix A, the following statements are equivalent A is singular det(A)=0 columns of A are linearly dependent rows of A are linearly dependent doesn’t exist Gaussian elimination has 0 rows either has no solution or infinite # of solutions Fundamental Matrix Definition of fundamental matrix Standard fundamental matrix Phase Plane analysis for 2nd order homogeneous D.E. Section 7.1 #3 Section 7.5 33 Section 7.6 #6 Section 7.7 #8 Section 7.9 #11 ...
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