alizpleyer
12.03.2020 •
Mathematics
1. For each equation in the following list, determine the set of initial conditions for which the Existence and Uniqueness Theorem guarantees a unique solution. (a) x? = kx (b) x? = sin x (c) x? = t / x (d) x? = cos t + x^3 ? x 2. Let x? = x^(1/3) (a) Show that the hypotheses of the Existence and uniqueness theorem do not hold in any box around the initial point (0, 0). (b) Using an argument similar to that for Torricelli�s law, construct a set of distinct solution curves that all satisfy the initial condition x(0) = 0. (c) Can you prove that the IVP x(0) = 1 has a unique solution? (d) Let x(t0) = x0 ?= 0. Show that the hypotheses of the Existence and uniqueness theorem are satsified if S is a sufficiently small box around (t0,x0). Conclude that the IVP has a unique solution as long as the solu- tion curve lies in S. Find the explicit form of the solution. 3. Let x? = x^2. Show that the hypotheses of the Existence and uniqueness theorem are satisfied for the initial condition x(0) = 1. Calculate the solution and determine the time of the singularity for the solution. Does this contradict the theorem? 4. Let x? =?sqrt(x) with x(0)=2. (a) Show that there exists a unique solution for this initial value problem. (b) Find an explicit solution for the IVP. ??(c) Notice that x1(t) = (sqrt(2) - t/2)^2 0<= t <= sqrt(8) x1(t) = 0 t>sqrt(8) and x2(t) = ( sqrt(2) -t/2 )^2 are two different functions that both seem to solve the differential equation and that both go through the initial point. Does the existence of these two functions imply that the IVP has more than one solution? Explain.
Solved
Show answers
More tips
- A Animals and plants Unraveling the Mystery of Loch Ness: What Does the Loch Ness Monster Look Like?...
- L Leisure and Entertainment Should You Buy a Ceramic Knife?...
- C Computers and Internet How to easily and quickly disable Firebug in Gmail and Google Docs...
- G Goods and services How to sew a ribbon: Tips for beginners...
- F Food and Cooking How to Make Mayonnaise at Home? Secrets of Homemade Mayonnaise...
- C Computers and Internet Which Phone is Best for Internet Surfing?...
- F Food and Cooking Everything You Need to Know About Pasta...
- C Computers and Internet How to Choose a Monitor?...
- H Horoscopes, Magic, Divination Where Did Tarot Cards Come From?...
Answers on questions: Mathematics
- M Mathematics Find the area of the composite shape lable your answer in square in...
- M Mathematics Find x y and z if x/9=y/6=z/3 and 2x+3y-4z =8...
- M Mathematics Solve for r. 9= lr + 4 Write your answers as integers or as proper or improper fractions in simplest form. or Submit...
- M Mathematics Explain please. Thank you...
- M Mathematics I need to know the value of X PLEASE HELP...
- M Mathematics This course is geometry can anyone help? I ll mark brainiest and please show how you did it...
- M Mathematics Summation notation HELP PLEASE...
- M Mathematics Figure A is a scale image of figure B What is the value of X...
- M Mathematics Find an equation of the line through 4,8 and parallel to y= 4x -6...
- M Mathematics I m stumped on the rest of the quiz I did the 1st five questions in part 1 of the Perimeter,circumference, and area quiz can you help with the rest...
Ответ: