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02 Comsider the linca ecomd order partia i (a) [2 marks] determine the class of the equation (b) [2 marks] Find the characteristics of the equation. 12 Marks] Sketch the characteristics in the...
Q2. Consider the equation (a) [2 marks] Find the characteristics of the equation. (b) [4 Marks] S...
Q2. Consider the equation (a) [2 marks] Find the characteristics of the equation. (b) [4 Marks] Sketch the characteristics in the (x,y) plane (c) [2 Marks] determine the characteristic coordinates (d) [6 marks] Reduce the equation to standard form and find its general solution (e) Use the general solution to find u(x, y), if it exists, for the following Cauchy data () [2 Marks] u(x,y)-2 on the curve y=x2 [2 Marks] u(x,y)-l on the curve y- (c) [2 Marks) u(x,y)-1...
MATH4474: Introduction to Partial Differential Equations Revision Exercises chapter 2, 4-6 Q1. Consider the linear second...
MATH4474: Introduction to Partial Differential Equations Revision Exercises chapter 2, 4-6 Q1. Consider the linear second order partial differential equation (i) (ii) (iii) Determine the class of this equation Find the characteristic coordinates Reduce the equation to canonical form 02. Consider the linear second order partial differential equation (a) [2 marks] determine the class of the equation (b) [2 marks] Find the characteristics of the equation. (c) 12 Marks] Sketch the characteristics in the (x, y) plane (d) [2 Marks]...
Q1. Consider the equation (a) Find the characteristics of the equation (2). (b) R educe the equ...
Q1. Consider the equation (a) Find the characteristics of the equation (2). (b) R educe the equation to standard form and find its general solution (c) Use the general solution to find ux, y), if it exists, for each of the following Cauchy data: (iii) u(x,y)-- on the curve 0(2) y n(x,y)-2e-y On the curve 1 x
Q1. Consider the equation (a) Find the characteristics of the equation (2).
(b) R educe the equation to standard form and find its...
2. (Undetermined Coefficients... In Reverse) Find a second order linear equation L(y) = f(0) with constant...
2. (Undetermined Coefficients... In Reverse) Find a second order linear equation L(y) = f(0) with constant coefficients whose general solution is: y=C et + Cell + tet (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation (h) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used...
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the ...
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
For the following ODEs, find roots λ of their characteristics equation and determine the type of damping
a) For the following ODEs, find roots λ of their characteristics equation and determine the type of damping (overdamped, undamped, underdamped, critically damped). For undamped and underdamped cases, find their angular frequency and period (14 points)i) y''+6 y'+10 y=0ii) 4 y''+8 y'+3 y=0iii) y''+8 y'+16 y=0b) Find the general solution for the following ODE 16 y''+8 y'+y=0(6 points)c) Find the particular solution of b) if y(0)=2 and y'(0)=2.5 .(5 points)
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)....
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2...
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively
15 pts] Sketch some representative...
(1 point) General Solution of a First Order Linear Differential Equation A first order linear differential...
(1 point) General Solution of a First Order Linear Differential Equation A first order linear differential equation is one that can be put in the form dy + P(2)y= Q(1) dz where P and Q are continuous functions on a given interval. This form is called the standard form and is readily solved by multiplying both sides of the equation by an integrating factor, I(2) = el P(z) da In this problem, we want to find the general solution of...
3. [20 pts.] (a) Find the equilibrium solutions of the equation y-υw-2)3. (b) Sketch the phase line of the equation...
3. [20 pts.] (a) Find the equilibrium solutions of the equation y-υw-2)3. (b) Sketch the phase line of the equation, and determine the stability of the equilibria you found in (a). (c) How does the solution with y(0) =-1 behave as t -» +00? How does the solution with y(0) 1 behave as t - --0?
3. [20 pts.] (a) Find the equilibrium solutions of the equation y-υw-2)3. (b) Sketch the phase line of the equation, and determine the stability...
Q2. Consider the equation (a) [2 marks] Find the characteristics of the equation. (b) [4 Marks] Sketch the characteristics in the (x,y) plane (c) [2 Marks] determine the characteristic coordinates (d) [6 marks] Reduce the equation to standard form and find its general solution (e) Use the general solution to find u(x, y), if it exists, for the following Cauchy data () [2 Marks] u(x,y)-2 on the curve y=x2 [2 Marks] u(x,y)-l on the curve y- (c) [2 Marks) u(x,y)-1...
MATH4474: Introduction to Partial Differential Equations Revision Exercises chapter 2, 4-6 Q1. Consider the linear second order partial differential equation (i) (ii) (iii) Determine the class of this equation Find the characteristic coordinates Reduce the equation to canonical form 02. Consider the linear second order partial differential equation (a) [2 marks] determine the class of the equation (b) [2 marks] Find the characteristics of the equation. (c) 12 Marks] Sketch the characteristics in the (x, y) plane (d) [2 Marks]...
Q1. Consider the equation (a) Find the characteristics of the equation (2). (b) R educe the equation to standard form and find its general solution (c) Use the general solution to find ux, y), if it exists, for each of the following Cauchy data: (iii) u(x,y)-- on the curve 0(2) y n(x,y)-2e-y On the curve 1 x
Q1. Consider the equation (a) Find the characteristics of the equation (2).
(b) R educe the equation to standard form and find its...
2. (Undetermined Coefficients... In Reverse) Find a second order linear equation L(y) = f(0) with constant coefficients whose general solution is: y=C et + Cell + tet (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation (h) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used...
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
a) For the following ODEs, find roots λ of their characteristics equation and determine the type of damping (overdamped, undamped, underdamped, critically damped). For undamped and underdamped cases, find their angular frequency and period (14 points)i) y''+6 y'+10 y=0ii) 4 y''+8 y'+3 y=0iii) y''+8 y'+16 y=0b) Find the general solution for the following ODE 16 y''+8 y'+y=0(6 points)c) Find the particular solution of b) if y(0)=2 and y'(0)=2.5 .(5 points)
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)....
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively
15 pts] Sketch some representative...
(1 point) General Solution of a First Order Linear Differential Equation A first order linear differential equation is one that can be put in the form dy + P(2)y= Q(1) dz where P and Q are continuous functions on a given interval. This form is called the standard form and is readily solved by multiplying both sides of the equation by an integrating factor, I(2) = el P(z) da In this problem, we want to find the general solution of...
3. [20 pts.] (a) Find the equilibrium solutions of the equation y-υw-2)3. (b) Sketch the phase line of the equation, and determine the stability of the equilibria you found in (a). (c) How does the solution with y(0) =-1 behave as t -» +00? How does the solution with y(0) 1 behave as t - --0?
3. [20 pts.] (a) Find the equilibrium solutions of the equation y-υw-2)3. (b) Sketch the phase line of the equation, and determine the stability...
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