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Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be...
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are i...
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0)...
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0) = x3 + 1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
Consider the partial differential equation, with the initial condition: 1 2 cup +3cºu, = 9x²y?, u(x,0)...
Consider the partial differential equation, with the initial condition: 1 2 cup +3cºu, = 9x²y?, u(x,0) = x3 +1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical...
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line.
4 Consider the autonomous differential equation y f(v)...
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...
Differential equation 1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+...
Differential equation
1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+4(x)-g(x) Subject to initial conditions y)oyy-Co) Consider the second order differential equation 2x2y" + 5xy, + y-r-x 2- The Existence of a Unique Solution Theorem says there will be a unique solution y(x) to the initial-value problem at x=而over any interval 1 for which the coefficient functions, ai (x) (0 S is n) and g(x) are continuous and a, (x)0. Are there any values of x for...
Consider the differential equation dy/dx = (y-1)/x.
Consider the differential equation dy/dx = (y-1)/x. (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (b) Let y = f (x) be the particular solution to the given differential equation with the initial condition f (3) = 2. Write an equation for the line tangent to the graph of y= f (x) at x = 3. Use the equation to approximate the value of f (3.3). (c) Find the particular solution y...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential equation linear? What is its order? Is it homogeneous? b) Try a solution of the form y=x". Is this a solution for some r? If so, find all such r. c) Based on your answer to a) about linearity and b) about what y=x" are solutions, make an educated guess a the general solution looks like. Try that guess and check that it works....
7. Consider the first order differential equation 2y + 3y = 0. (a) Find the general...
7. Consider the first order differential equation 2y + 3y = 0. (a) Find the general solution to the first order differential equation using either separation of variables or an integrating factor. (b) Write out the auxiliary equation for the differential equation and use the methods of Section 4.2/4.3 to find the general solution. (c) Find the solution to the initial value problem 2y + 3y = 0, y(0) = 4.
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a)...
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a) Check that y(t) = 1-1 is a solution to this equation. (b) Find another solution (t) such that yı(t) and (t) are linearly independent (that is, wit) and y(t) form a fundamental set of solutions for the differential equation).
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0) = x3 + 1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
Consider the partial differential equation, with the initial condition: 1 2 cup +3cºu, = 9x²y?, u(x,0) = x3 +1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line.
4 Consider the autonomous differential equation y f(v)...
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...
Differential equation
1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+4(x)-g(x) Subject to initial conditions y)oyy-Co) Consider the second order differential equation 2x2y" + 5xy, + y-r-x 2- The Existence of a Unique Solution Theorem says there will be a unique solution y(x) to the initial-value problem at x=而over any interval 1 for which the coefficient functions, ai (x) (0 S is n) and g(x) are continuous and a, (x)0. Are there any values of x for...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential equation linear? What is its order? Is it homogeneous? b) Try a solution of the form y=x". Is this a solution for some r? If so, find all such r. c) Based on your answer to a) about linearity and b) about what y=x" are solutions, make an educated guess a the general solution looks like. Try that guess and check that it works....
7. Consider the first order differential equation 2y + 3y = 0. (a) Find the general solution to the first order differential equation using either separation of variables or an integrating factor. (b) Write out the auxiliary equation for the differential equation and use the methods of Section 4.2/4.3 to find the general solution. (c) Find the solution to the initial value problem 2y + 3y = 0, y(0) = 4.
(3) Consider the differential equation ty' + 3ty + y = 0, 1 > 0. (a) Check that y(t) = 1-1 is a solution to this equation. (b) Find another solution (t) such that yı(t) and (t) are linearly independent (that is, wit) and y(t) form a fundamental set of solutions for the differential equation).
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