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Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r R is a constant. (a) Find the value(s) of...
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any...
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any constant solutions of this differential equation and hence write down the solution with initial value y=- when r=7 (b) Use partial fractions to evaluate 1 dy. 1 (c) Use the method for solving separable differential equations to solve this DE in the case where y 0 when r T. You may assume that the solution does not cross the constant solutions you found in...
Consider the nondimensional differential equation du where u is an unknown parameter (constant) (a) Determine the equil...
Consider the nondimensional differential equation du where u is an unknown parameter (constant) (a) Determine the equilibrium solutions in terms of μ. (b) Draw the bifurcation diagram and clearly identify the bifurcation point. (c) Classify the stability of the branches in your bifurcation diagram using the process in class where we assumed u(t)uilibrium +u(t) where uequilibrium is the constant(s) you determined in (a) Repeat the steps in exercise (2) for the nondimensional differential equation given by du_2 dt where u...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the s...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
Consider the differential equation (1 2 yay 0, where a E R is a constant. (a)...
Consider the differential equation (1 2 yay 0, where a E R is a constant. (a) By analysing the equation, show that there are two linearly independent power series solutions in powers of for la<1 (b) Find two linearly independent solutions. Note: The recurrence relation you derive should be the following (or equivalent to it) (n-a)(n a) an (n1)(n2) n 2 0. an+2 polynomial solution of (c) Show that if a is (nonnegative) integer n, then there is a degree...
Question 2: (5+15=20 points) a) Find the value of the constant k e R for which...
Question 2: (5+15=20 points) a) Find the value of the constant k e R for which the differential equation (2+ y + xy) dx + (1+2+kx*y) dy=0 is exact. b) Find the solution of the initial value problem using the value of k you found in part (a). (2+ y + r?y?)dr + (1 + x + k.xºy)dy = 0, y(0) = 2
1. Consider the differential equation" = y2 - 4y - 5. a) Find any equilibrium solution(s)....
1. Consider the differential equation" = y2 - 4y - 5. a) Find any equilibrium solution(s). b) Create an appropriate table of values and then sketch (using the grid provided) a direction field for this differential equation on OSIS 3. Be sure to label values on your axes. c) Using the direction field, describe in detail the behavior of y ast approaches infinity. 2. Short answer: State whether or not the differential equation is linear. If it is linear, state...
4. Consider the differential equation N'-200,000N +210ON2 N3. a. Find the three equilibrium points of the differential equation and determine their stability using the graphical qualitative metho...
4. Consider the differential equation N'-200,000N +210ON2 N3. a. Find the three equilibrium points of the differential equation and determine their stability using the graphical qualitative method. b. Determine the long-term behavior of the differential equation and explain, in biological terms, what happens to a population modeled by the differential equation for different initial populations, No (8 points) 6 points)
4. Consider the differential equation N'-200,000N +210ON2 N3. a. Find the three equilibrium points of the differential equation and determine...
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r...
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r, θ) (i) (7 marks] if f (θ)-sin|-θ | , find the specific solution u (r,0) (ii) [ (ii) [5 marks] Use the solution in (ii) to deduce that 4n1-9) 18 Q4. [24marks] Consider the second order linear partial differential equation
Q3. [22 marks] The Dirichlet's...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant....
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questio...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questions (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: {stable / saddle / unstable(not saddle)) (c) Show that there is no periodic solution in a simply connected region (Hint: Use the corollary to Theorem 11.5.1)
Consider the nonlinear second-order differential equation where...
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any constant solutions of this differential equation and hence write down the solution with initial value y=- when r=7 (b) Use partial fractions to evaluate 1 dy. 1 (c) Use the method for solving separable differential equations to solve this DE in the case where y 0 when r T. You may assume that the solution does not cross the constant solutions you found in...
Consider the nondimensional differential equation du where u is an unknown parameter (constant) (a) Determine the equilibrium solutions in terms of μ. (b) Draw the bifurcation diagram and clearly identify the bifurcation point. (c) Classify the stability of the branches in your bifurcation diagram using the process in class where we assumed u(t)uilibrium +u(t) where uequilibrium is the constant(s) you determined in (a) Repeat the steps in exercise (2) for the nondimensional differential equation given by du_2 dt where u...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
Consider the differential equation (1 2 yay 0, where a E R is a constant. (a) By analysing the equation, show that there are two linearly independent power series solutions in powers of for la<1 (b) Find two linearly independent solutions. Note: The recurrence relation you derive should be the following (or equivalent to it) (n-a)(n a) an (n1)(n2) n 2 0. an+2 polynomial solution of (c) Show that if a is (nonnegative) integer n, then there is a degree...
Question 2: (5+15=20 points) a) Find the value of the constant k e R for which the differential equation (2+ y + xy) dx + (1+2+kx*y) dy=0 is exact. b) Find the solution of the initial value problem using the value of k you found in part (a). (2+ y + r?y?)dr + (1 + x + k.xºy)dy = 0, y(0) = 2
1. Consider the differential equation" = y2 - 4y - 5. a) Find any equilibrium solution(s). b) Create an appropriate table of values and then sketch (using the grid provided) a direction field for this differential equation on OSIS 3. Be sure to label values on your axes. c) Using the direction field, describe in detail the behavior of y ast approaches infinity. 2. Short answer: State whether or not the differential equation is linear. If it is linear, state...
4. Consider the differential equation N'-200,000N +210ON2 N3. a. Find the three equilibrium points of the differential equation and determine their stability using the graphical qualitative method. b. Determine the long-term behavior of the differential equation and explain, in biological terms, what happens to a population modeled by the differential equation for different initial populations, No (8 points) 6 points)
4. Consider the differential equation N'-200,000N +210ON2 N3. a. Find the three equilibrium points of the differential equation and determine...
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r, θ) (i) (7 marks] if f (θ)-sin|-θ | , find the specific solution u (r,0) (ii) [ (ii) [5 marks] Use the solution in (ii) to deduce that 4n1-9) 18 Q4. [24marks] Consider the second order linear partial differential equation
Q3. [22 marks] The Dirichlet's...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questions (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: {stable / saddle / unstable(not saddle)) (c) Show that there is no periodic solution in a simply connected region (Hint: Use the corollary to Theorem 11.5.1)
Consider the nonlinear second-order differential equation where...
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