Understand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.

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1 Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics Know how to classify differential equations terms you should know include: Order, Linear, Nonlinear, Implicit Solution, Initial Value Problem, Direction Field, Phase Portrait, Exact, Seperable For first order systems, you should know how to roughly sketch a direction field and identify the direction field that goes with a particular equation. From a direction field you should be able to identify the behavior of a solution given an initial condition. Know how to check to see if a particular function satisfies a differential equation or not.. First Order Systems Know how to solve first order linear differential equations and IVP s using integrating factors. Know how to solve first order differential equations and IVP s using separation of variables. Be able to recognize when an implicit solution can be solved to find an explicit solution. Know how to solve first order EXACT differential equations and IVP s. Be able to recognize when an implicit solution can be solved to find an explicit solution. Understand how to determine if a first order differential equation has a unique solution and in the case of first order linear equations know how to determine an interval on which that solution exists. Don t forget that the solution may exist on a bigger interval, but you can determine an interval on which it must exist in the case of a linear first order IVP.. Second Order Equations Know how to show that two solutions are dependent or linearly independent using the Wronskian. Know how to compute the general solution to any homogeneous second order differential equation. This involves looking at the characteristic equation and its roots, which are either both real and different, real and repeated, or a complex conjugate pair. Know how to compute a particular solution to a non-homogeneous second order differential equations. This involves either method of undetermined coefficients and/or variation of parameters. Also recall that variation of parameters can be used for non-constant coefficient problems as well. Knowing the particular and the homogeneous solution, know how to compute the solution to an IVP. Know how to set up a second order differential equation for mechanical vibration problems. For these problems you should know how to write your solutions in the form Re αt cosωt δ. Know how to compute the bounding curves for decaying solutions. Know how to compute, period, pseudo-period, Amplitude, frequency, pseudo-frequency. Know how to tell if resonance occurs or not in forced systems. Be able to determine is a system is critically damped or over damped. 4. Higher Order Linear Equations. Know how to compute the solutions to higher order linear differential equations from using the roots of the characteristic equation. This might involve long division to factor the characteristic polynomial. Be able to use variation of parameters and undetermined coefficients to find particular solutions for non-homogeneous higher order linear ODE s.

2 Understand the existence and uniqueness theorems and what they tell you about solutions to initial value problems. 5. Series Solutions Know how to find singular points of a differential equation. Know how to determine a minimum for the radius of convergence of a series solution around a given ordinary point. Know how to determine the recurrence relation for a series solution. Know how to use the initial conditions to determine the first few terms of a series solution and then use the recurrence relation to find the later terms using the first few terms. 6. Laplace Transforms Know how to compute the Laplace Transform from using both the integral definition and using the table. Know how to compute the Inverse Laplace Transform of a function using the table. Given a differential equation and initial conditions know how to determine the Laplace Transform of the solution to the equation. Then you have to use the inverse transform to get the actual solution. Often this involves partial fraction decomposition. Know how the Laplace Transform and Inverse Laplace Transforms are used to handle step functions and impulse functions and be able to use them in solving ODE s. 7. First Order Systems For a constant matrix A, know how to find the general solution to the first order linear system x = A x, This involves knowing how to compute eigenvalues and eigenvectors and how to deal with complex and repeated eigenvalues. Also be able to describe the general behavior of the solutions to these systems given initial values what happens to solutions as t? For systems be able to describe what the phase portrait looks like and classify any equilibirum points as a saddle point, center, source, sink, improper node, spiral point. Be able to determine if these points are stable, asymptotically stable or unstable. Know how to show vector solutions are independent using the Wronskian. Know how to use undetermined coefficients and variation of parameters to solve a non-homogeneous system x = A x + gt, make sure you know what to do if you guess looks like part of your homogeneous solution. Be able to determine the solution to an initial value problem for a system by solving for unknown constants. Know how to rewrite a linear differential equation as a first order system. 8. Numerical Methods Know how to carry out a few steps of either Euler, Improved Euler or RK4. Know the difference between local and global error. Know the order of both kinds of error for each of the three methods and how to use this error estimates to approximate the change in error when the number of steps increases or approximate the step size or number of steps needed to achieve a certain error bound.

3 Problems. Series Solutions Near Ordinary Points a Consider the following differential equation: x xy + xy + y =0 What are the singular points for this differential equation? First we rewrite the equation by dividing by x x, this gives: Thus the singular points are at x =0and x =. b If yx = y + x y + xx y =0. a n x / n is a series solution to the differential equation stated in a, determine a lower bound for the radius of convergence of this series. The closest singular point to x =/is x =0and / 0 =, thus the radius of convergence is at least / and a nx / n converges for at least x / < / or for 0 <x<.

4 c Consider the following initial value problem: 4 x y +y =0, y0 =, y 0 = This differential equation has an ordinary point at x =0. Consequently there exists a series solution of the form yx = b n x n. Determine the first five nonzero terms of this series solution. We begin by differentiating the series, y = y = y = b n x n nb n x n n= nn b n x n = n + n + b n+ x n n= Substituting these into the equation gives: Seperating some terms we have: 4 x nn b n x n + b n x n =0 n= n= 4n + n + b n+ x n nn b n x n + b n x n =0 Only the first and the last sum has constant and linear terms, so we will pull those out and then combine the remaining terms into one sum. 8b + 4b x +b 0 +b x + [4n + n + b n+ + nn b n x n =0 n= Now the initial condition y0 = tells us that b 0 =and y 0 =, tells us that b =. 4

5 The recurrence relationship is: 8b +b 0 = 0 4b +b = 0 4n + n + b n+ + nn b n = 0, n =,, 4,... Solving these equations from the previous we have: Thus we have that the first five terms are: 8b = b 0 = b = 4 4b = b = 4 b = 44b 4 + b =0 b 4 =0 454b 5 + b =0 b 5 = 0 y = x 4 x + x + 0 x

6 . Find the general solution of the following system of equations: x = x +y y = 5x y Sketch a few trajectories and in complete sentences describe the behavior of the solutions as t. To begin solving this system we must first find the eigenvalues and corresponding eigenvectors of the matrix A =. First we find the roots of the characteristic polynomial given by deta ri. In 5 this case deta ri = r r + 0 = r +9. The corresponding roots are r = ±i. Since we are in the case of complex eigenvalues λ =0, µ = we actually only need to build the eigenvector for one of the eigenvalues, let us do this for r =i. We must then solve the system A ii ξ =0. Setting up the augmented system we have: If the entries of ξ = [ [ [ i 0 i 0 +i i 0 5 i 0 5 i 0 [ +i [ [ 0 0 +i 0 i +5 +i i ξ ξ then we must have ξ i ξ =0or ξ = i ξ. Thus, i ξ = k 5 It is possible that you might get other equivalent forms for ξ. If we choose k =and write ξ in terms of its real an imaginary parts ξ = a + i b one has a = and 5 b =. Using the formulas from class we 0 have that two real valued independent solutions are then given by: [ ut = e λt a cosµt b sinµt = [ vt = e λt a sinµt+ b cosµt = cost + sint 5 cost sint cost 5 sint Note: You may get alternate equivalent solutions if you choose a different eigenvector. The resulting general solution will be: cost + sint sint cost xt =c + c 5 cost 5 sint Depending on your choice of eigenvectors another equivalent solution would be: cost sint xt =c + c cost + sint sint cost continued on the next page... 6

7 Based on the fact that there are purely imaginary eigenvalues of the constant matrix A and that multiplying 0 the matrix A = we know that the equilibrium point at 0, 0 is a stable but not asymptotically stable center, with trajectories that rotate clockwise. As t, the trajectories continue to oscillate around the center. Below is a sketch of the phase portrait with some trajectories drawn in. " # $ % & Y #!!&!%!$!#!"!"!#!$!%!&! & % $ # " x. Find the solution to the system x = x + y y = x y With the initial conditions x0 = and y0 =. 7

8 To begin solving this system we must first find the eigenvalues and corresponding eigenvectors of the matrix A =. First we find the roots of the characteristic polynomial given by deta ri. In this case deta ri = r r+ 9 4 = r r + 9/4 =r r +/4 = r /. Thus there is one repeated root, r =/. To find the eigenvector that is associated to the eigenvalue we must solve the system A /I ξ =0. Setting up the augmented system we have: [ [ 0 [ If the entries of ξ = This gives us one solution ξ ξ then we must have ξ + ξ =0or ξ = ξ. Thus, ξ = k x = e t Since the eigenspace corresponding to the eigenvalue r = / is less then the multiplicity, then we have to build our second solution in the form: x = te t + νe t, where ν is the solution to the matrix equation A Iν = ξ, or in this case we must solve: ν = ν Our augmented system is then: [ [ [ Consequently we must have that ν + ν = or ν = ν. Giving: k ν = k for any value k. For ease choose k to be 0 giving, ν = 0 The resulting general solution will be: x = te t 0 +. Thus our second solution is then: e t, [ xt =c e t + c te t 0 + e t However, this is not the entire answer we still need to find c and c so that the initial conditions are 8 satisfied. Continued on the next page...

9 Before I compute c and c I will note that this is an unstable improper node and the phase portrait with some trajectories drawn in is shown below:!"#"$"%"&"'""! &"#"$"'!"#"$"% * + ' %! $% $ $' $+ $* $* $+ $' $ $% % ' + * & Now to find c and c we evaluate our solution at t =0and set it equal to 0 = c + c This row implies that c =and then we have = + c from the second row. c =/The resulting solution will be: xt = e t + [ te t 0 + e t 9

10 4. Find the general solution to the following system using variation of parameters x = 4x +y + t y = x y + t +4 To begin solving this system we must first find the eigenvalues and corresponding eigenvectors of the 4 matrix A =. First we find the roots of the characteristic polynomial given by deta ri. In this case deta ri = 4 r r 4=r +5r +4 4=r +5r = rr + 5. The two eigenvalues are 0 and 5. To find the eigenvector that is associated to the eigenvalue r =0, we must solve the system A 0I ξ =0. Setting up the augmented system we have: [ [ If the entries of ξ = ξ ξ then we must have 4ξ +ξ =0or ξ = ξ. Thus, ξ = k Choosing k =, this gives us one solution to the homogeneous problem, x = e 0t = To find the eigenvector that is associated to the eigenvalue r = 5, we must solve the system A+5I ξ =0. Setting up the augmented system we have: If the entries of ξ = [ [ ξ ξ then we must have ξ +ξ =0or ξ = ξ. Thus, ξ = k Choosing k =, this gives us our second solution to the homogeneous problem, x = e 5t Consequently we have the fundamental matrix for the homogeneous problem: e 5t Ψt = e 5t The determinant of the fundamental matrix is e 5t +4r 5t =5e 5t. 0

11 Consequently the inverse of the fundamental matrix is given by: Ψ t = e5t 5 e 5t e 5t = e5t 5 e5t Using variation of parameters tells us that the general solution is of the form: x =Ψ Ψ tgtdt + Ψt c, for some arbitrary constant vector c. To plug in the values we have we will do this piece by piece. First let s compute the product Ψ tgt. Ψ tgt = 5 5 t 5 e5t 5 e5t t = 5 t t t +4 5 e5t t + 5 e5t t + 4 = e5t 4 5 e5t Now computing the integral: Ψ tgtdt = 5 t 5 e5t t e5t Now computing the product: Ψt Ψ e 5t lnt+ 8 lnt+ 8 tgtdt = e 5t 4 = Putting this all together we have: xt = for some unknown vector c. If c = xt = lnt+ c c lnt+ 8/5 t + 6/5 then we have that: 8/5 t + 6/5 lnt+ 8 dt = 5 t e5t lnt+ 6 5 t t = lnt+ 8/5 e 5t + 4/5 e 5t c 8/5 e 5t + c 4/5 + c e 5t 8/5 8/5 t+ 6/5 4/5

12 5. Series Solutions Near Ordinary Points a Consider the following initial value problem: 4 x y +y =0, y = 0, y = This differential equation has an ordinary point at x =0. Consequently there exists a series solution of the form yx = b n x + n. Determine the first five nonzero terms of this series solution. We begin by differentiating the series, y = y = y = b n x + n nb n x + n = n + b n+ x + n n= nn b n x + n = n + nb n+ x + n = n + n + b n+ x + n n= n= Before substituting these into the equations first rewrite the equation by replacing x with x+ giving: 4 x + y +y = 0 Expanding we have: 4 x + x + + y +y = 0 or x + + x + y +y = 0 or y x + y + x + y +y = 0 Now substituting in we have: or n + n + b n+ x + n x + nn b n x + n n= +x + n + nb n+ x + n + b n x + n = 0 n= n + n + b n+ x + n nn b n x + n n= + n + nb n+ x + n + b n x + n = 0 n=

13 Only the first and the last sum have constant and linear terms and the third term has a linear term, so we will pull those out and then combine the remaining terms into one sum. + 6b + 8b x + + 4b x + + b 0 +b x + [n + n + b n+ nn b n n + nb n+ +b n x n = 0 n= Simplifying slightly gives: + 6b +b 0 + 8b +4b +b x + [n + n + b n+ + nn + b n n + nb n+ x n = 0 n= Now the initial condition y = 0 tells us that b 0 =0and y =, tells us that b =. The recurrence relationship is: 6b +b 0 = 0 8b +4b +b = 0 n + n + b n+ + nn b n n + nb n+ = 0, n =,, 4,... Solving these equations from the previous we have: Thus we have that the first five terms are: 6b = b 0 =0 b =0 8b = b 4b = 4 b = 9 4b 4 + b b =0 b 4 = 7 54b 5 + b 4b 4 =0 b 5 = 45 y = x + 9 x x x