<p> MAT 361: What do I need to know for the final?</p><p>1. Types of Equations a. First Order Equations b. Exact and Inexact Forms c. Second Order Linear Constant Coefficient d. Cauchy-Euler e. Forced Harmonic Oscillators – know terminology and behavior! f. Resonant Systems g. Systems of Equations 2. Methods a. Reduction of Order b. Method of Undetermined Coefficients c. Method of Variation of Parameters d. Numerical Solution – Euler’s Method e. Laplace Transform f. Inverse Laplace Transform g. Power Series Method h. Frobenius Method i. Matrix Methods – Eigenvalue Problems 3. Applications a. Free Fall b. Growth and Decay i. Populations ii. Radioactivity – Half Life c. Newton’s Law of Cooling d. Mixing Problems i. Rate of Change of Quantity = Rate In – Rate Out e. Oscillations (SHM, damped, forced) 4. Specifics a. First Order Differential Equations i. Separation of Variables dy dy 1. = f( t ) g ( y )� f ( t ) dt dt蝌 g( y ) 2. General Solutions – Implicit and Explicit 3. Initial Value Problems – Particular Solutions ii. Linear Differential Equations 1. Find integrating factors and solve initial value problems a. y'+ a ( x ) y = f ( x ) x m(x )= exp a ( x ) dx� ( m y) ' m f</p><p>1 轾x y( x )=犏m ( t ) f ( t ) dt + C m(x ) 臌 dy iii. Autonomous Equations = f( y ) dt</p><p>1. Equilibrium solutions f( y0 )= 0. 2. Classification (stable, unstable) 3. Bifurcations b. ay'' by '  cy  0 i. Solve characteristic equation ar2  br  c  0. ii. Three Cases: </p><p> r1 t r 2 t 1. k1 e k 2 e rt 2. e( k1 k 2 t ) t 3. e[ k1 cos( t ) k 2 sin(  t )] , r  i . c. ay'' by '  cy  f ( t ).</p><p> i. y( t ) yh ( t )  y p ( t ) n at n at ii. f( t ) t e  yp  ( A n t ⋯  A0 ) e</p><p> iii. f( t ) sin( at )  yp  A sin( at )  B cos( at ) iv. If f( t ) term is solution of homogeneous equation, multiply guess by additional powers of t. v. Variation of Parameters f( x ) c' y+ c ' y = 0, c' y '+ c ' y ' = 1 1 2 2 1 1 2 2 a( x ) d. ax2 y'' bxy '  cy  0, y  y ( x ) i. Solve characteristic equation ar( r 1)  br  c  0. ii. Three Cases: </p><p> r1 r 2 1. k1 x k 2 x r 2. x( k1 k 2 ln | x |)  3. x[ k1 cos( ln | x |) k 2 sin(  ln | x |)] , r  i . e. Laplace Transform  i. Definition Y( s ) L y ( t )   y ( t ) est dt . 0 ii. Particular Functions tn, e at ,sin at ,cos at , H ( t- a ),d ( t - a ) iii. Differential Equations 2 dy  d y  2 L   sY( s )  y (0) , L   s Y( s )  sy (0)  y '(0) dt  dt 2  t iv. Convolution (f g )( t )  f ( t  u ) g ( u ) du . 0 v. Convolution Theorem L[ f* g] = F ( s ) G ( s ) or L-1 [ F( s ) G ( s )] = f * g f. Systems of Equations i. Phase Plane and Phase Portraits 1. Vector Fields 2. Direction Fields 3. Equilibrium Solutions 4. y vs x and x(t), y(t) plots ii. Eigenvalue Problems – Solve for eigenvalues and eigenfunctions. iii. Solution of systems iv. Bifurcations v. Finding and Classifying Equilibrium Solutions in Nonlinear Systems g. Power Series Methods i. Identify ordinary, regular singular, and irregular singular points 1  f(n ) ( a ) ii. Taylor Series for ex ,sin( x ),cos( x ), , f( x ) ( x  a )n 1 x n0 n! iii. For Ordinary Points  n 2 1. y( x ) cn ( x  x0 )  c 0  c 1 ( x  x 0 )  c 2 ( x  x 0 )  n0 2. Get recursion formula</p><p>3. Solve for cn 's 4. Sum series iv. For Singular Points  n r 1. y( x ) cn ( x  x0 ) n0 2. Get indicial equation for r 3. Get recursion formula</p><p>4. Solve for cn 's 5. Sum series</p>