Math 267, Section E2, 2009f Hailiang Liu December 4, 2009
Chapter 1: First Order Equations pp1-99 Chapter 2: Linear equations of higher order, pp100-193 Chapter 3: Power Series Methods pp194- Chapter 4: Laplace Transform methods pp266– 325 Chapter 5: Linear Systems of Differential Equations pp 326–429 Chapter 6: *Numerical Methods pp 430–479 Chapter 7: Nonlinear Systems and Phenomena pp480–554
Schedule reminder ◮ Review on your own on Dec. 7, 8, 10 ◮ You may ask questions these days: 10:00am–11:00am Dr. Jun PAN, Carver 384 ◮ Last review session will be in classroom on Dec. 11; 12:10pm–1:00pm ◮ Final is to be held on Dec. 15, 12:00pm–2:00pm. ◮ Final involves total 8 problems, 100 points, 120 minutes.
Chapter 1: Main topics dy dx = f ( x , y ) , or M ( x , y ) dx + N ( x , y ) dy = 0 . ◮ Types of equations Integrable equation Separable equation Linear equation Exact equation ◮ Methods to solve each type ... ◮ Substitution methods and equation reduction.
Integrable equations ◮ Mathematical form dy dx = f ( x ) Its solutions is � y = f ( x ) dx + C . ◮ Remember to add an integral constant. ◮ Review of how to integrate a given function.
Separable equations ◮ Mathematical form dy dx = f ( x ) g ( y ) Its solutions is � dy � g ( y ) = f ( x ) dx + C . ◮ Be sure no loss of solutions when separating variables
Linear equation dy dx + P ( x ) y = Q ( x ) ◮ Integrating factor R P ( x ) dx ρ ( x ) = e ◮ Multiply the equation by ρ to obtain d dx [ ρ y ] = Q ( x ) ρ ( x ) ◮ Integration gives �� � y = ρ − 1 Q ( x ) ρ ( x ) dx + C .
Exact equation M ( x , y ) dx + N ( x , y ) dy = 0 . ◮ Definition: the equation is exact if there exists an F such that M ( x , y ) dx + N ( x , y ) dy = d ( F ( x , y )) , Its solutions is F ( x , y ) = C . ◮ Criterion: If M y = N x , then the equation must be exact. ◮ How to find F: i) Regroup terms and use the product rule to observe F ; ii) Integrate F x = M to obtain � F = Mdx + g ( y ) Mdx + dg Then N = F y = ∂ � dy . ∂ y
Substitution methods ◮ y ′ = F ( ax + by + c ) , v = ax + by + c , then dv dx = a + bF ( v ) . ◮ Homogeneous equation � y y ′ = F � , v = y / x . x ◮ The Bernoulli equation: y ′ + P ( x ) y = Q ( x ) y n , v = y 1 − n .
Reducible equations ◮ no explicit ’y’ F ( x , y ′ , y ′′ ) = 0 then p = y ′ → F ( x , p , p ′ ) = 0 . ◮ No explicit ’x’ F ( y , y ′ , y ′′ ) = 0 then p = y ′ → F ( y , p , p dp dy ) = 0 .
Exam 1– 9/14/09 1. (5 points) Determine the order of the given differential equations. (a) d 2 y dt 2 + sin( t + y ) = sin t dt ) 2 + ty 3 = 0 (b) ( dy 2. (5 points) Is the function y = 3 t 2 a solution of the differential equation ty ′ − y = t 2 ? Verify your answer. 3. (10 points) Find the general solution of the equation y ′ = 4 t − 6, and then find the particular solution with the initial condition y (0) = 1. 4. (15 points) Find the value of a for which the equation (2 xy 2 + ay ) + (2 x 2 y + 2 x ) y ′ = 0 is exact, and then solve it using that value of a . 5. (15 points) What type is the equation x 2 y ′ + 2 xy = 5 y 3 ? Find its general solution.
Chapter 2: Main topics Ly := y ( n ) + p 1 ( x ) y ( n − 1) + · · · + p n − 1 ( x ) y ′ + p n ( x ) y ◮ nth order homogeneous linear equations Ly = 0 and solution structures y c = c 1 y 1 + · · · c n y n . ◮ Characteristic equation method y = e rx for finding y i ( x ) ◮ Nonhomogeneous euqations Ly = f ( x ) and solution structures y = y c ( x ) + y p ( x ) . ◮ How to find y p for p i constants and f special.
Solution structures ◮ Superposition of solutions L [ ay 1 + by 2 ] = aL [ y 1 ] + bL [ y 2 ] = 0 . ◮ Linear independence and Wronskians W = W ( y 1 , · · · y n ) ◮ Gneeral solution of homogeneous linear equations y c = c 1 y 2 + · · · c n y n . ◮ General solution of non-homogeneous solution y = y c ( x ) + y p ( x ) .
Find y c for Ly = 0 ◮ Write down the characteristic equation r n + p 1 r n − 1 + · · · p n − 1 r + p n = 0 . ◮ r is real, so y = e rx is a solution ◮ r is complex, so two solutions are y 1 = Re [ e rx ] , y 2 = Im [ e rx ] . ◮ If r repeated of multiplicity k , then y 1 = e rx , y 2 = xe rx , · · · y k = x k − 1 e rx .
Find y p by method of underdetermined coefficients ◮ The method is valid only if p i are constants and f is either a polynomial in x ; an exponential e rx ; coskx or sinkx , or linear combinations of them. ◮ Try solution of the same form y p = x s ˜ f ( x ) . Here s is chosen so that no term in this solution appears in y c . ◮ Method of variation of parameters y p = u 1 ( x ) y 1 + u 2 ( x ) y 2 and then try to find u i .
Exam 2– 10/02/09 1. [10 points.] Show that y 1 ( x ) = e − 4 x and y 2 ( x ) = xe − 4 x are two linearly independent solutions for y ′′ + 8 y ′ + 16 y = 0 , then find a solution satisfying y (0) = 2 and y ′ (0) = − 1. 2. [10 points.] Find the general solution of y ′′′ − y ′′ + y ′ − y = 0 . 3. [10 points.] Consider the following linear, non-homogeneous equation, y ′′ + 4 y = f ( x ) . For each of the following cases, write and appropriate form of a particular solution (you do not need to find the exact solution.) a. f ( x ) = x 2 e 2 x b. f ( x ) = x cos(2 x ) c. f ( x ) = x − 1 + e x sin(2 x ) 4. [10 points.] Find the solution to the following initial value problem, y ′ (0) = 15 y ′′ − y = e x , y (0) = 3 , 2 .
Chapter 3: Main topics ◮ What is a power series, its convergence radius ◮ Series solution near ordinary points ∞ � c n ( x − a ) n . y = n =0 ◮ The radius if convergence= at least as large as the distance from a to the nearest singular point of the equation. ◮ Steps to find c n — two types of recurrence relations Two-term recurrence More-term recurrence
Chapter 4: Main topics ◮ Laplace transform of typical functions and their inverse ◮ Properties of Laplace transform ◮ Transformation of Initial value problems ◮ Translation and partial fractions ◮ Derivatives, integrals, products of transforms ◮ Piecewise continuous ◮ Impulse and Delta functions
Laplace Transform � ∞ 0 e − st f ( t ) dt , ◮ F ( s ) = L { f ( t ) } = s > a . ◮ Laplace of { 1 , e at , t a , t n , coskt , sinkt , u ( t − a ) } =? ◮ Any linear combinations can be transformed by using L { af + bg } = aL { f } + bL { g } . ◮ Inverse Transforms f ( t ) = L − 1 { F ( s ) } . ◮ Existence and Unique Theorem: If f is piecewise continuous and | f | ≤ Me ct , then F ( s ) exists for all s > c and is unique. Moreover, s →∞ F ( s ) = 0 . lim
Transform of IVPs ◮ Transform of derivatives L { f ′′ ( t ) } = s 2 F ( s ) − sf (0) − f ′ (0) L { f ′ ( t ) } = sF ( s ) − f (0) , ◮ Transform of ax ′′ + bx ′ + cx = f ( t ) with x (0) = x 0 , x ′ (0) = x 1 is L { x } = F ( s ) + a ( sx 0 + x 1 ) + bx 0 . as 2 + bs + c ◮ Transform of Integrals �� t � = F ( s ) L f ( τ ) d τ . s 0
Translation and partial fractions ◮ Translation in s L { e at f ( t ) } = F ( s − a ) . ◮ Transform of e at { t n , coskt , sinkt } ◮ Decomposition of a fraction A 2 + B 2 s A 1 P ( s ) / Q ( s ) = s − a + ( s − a ) 2 + b 2 + ... ◮ A practice example: solve x ′′ + 4 x ′ + 4 x = t 2 ; x (0) = x ′ (0) = 0 .
Useful Transform formulas ◮ Convolution L { f ∗ g } = L { f } · L { g } . ◮ Differentiation of Transforms L { t n f ( t ) } = ( − 1) n F ( n ) ( s ) . ◮ Integration of transforms � ∞ � f ( t ) � L = F ( τ ) d τ. t s ◮ Three examples: find � 2 � � sinht � ( i ) L − 1 ( ii ) L − 1 { tan − 1 (1 / s ) } ( iii ) L ( s − 1)( s 2 + 4) t
Transform of special functions ◮ Piecewise continuous L { u ( t − a ) f ( t − a ) } = e − as F ( s ) . ◮ Periodic function* f ( t + p ) = f ( t ) � p 1 e − st f ( t ) dt . F ( s ) = 1 − e − ps 0 ◮ Pulse L { δ ( t − a ) } = e − as . ◮ Practices: � e − as � L − 1 s 3 x ′′ + 4 x = δ ( t − 2 π ); x (0) = 1 , x ′ (0) = 1 .
Exam 3– 10/22/09 1. [10 points.] Using power series methods to solve the equation y ′′ + xy = 0 . 2. [10 points.] Show that the equation xy ′ + y = 0 . has no non-trivial power series solution of the form y = � ∞ n =0 c n x n . 3. [10 points.] Apply the definition to find the Laplace transform of the function e kt ; then use the obtained result and the relation coskt = e ikt + e − ikt to find L{ coskt } . 2 4. [10 points.] Using Laplace transform to solve the initial value problem x ′′ − x = 5; x (0) = x ′ (0) = 0 . 5. [10 points.] Using the formula L{ tf ( t ) } = − F ′ ( s ) to find L − 1 { ln ( s − 1) } , s > 1 .
Chapter 5: Main topics ◮ How to rewrite a higher equations into a first order system; ◮ The method of elimination (to reduce to higher order equation to solve) ◮ Solution structure of linear systems ˙ x = P ( t ) x + f ( t ): x = c 1 x 1 + · · · c n x n + x p ( t ) . ◮ Eigenvalue method x = ve λ t ◮ Matrix exponentials e At ◮ Nonhomogeneous system ˙ x = Ax + f ( t ).
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