(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
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Q1. Consider the equation (a) Find the characteristics of the equation (2). (b) R educe the equ...
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]...
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...
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 (x, y) plane (c) (d) [2 Marks] determine the characteristic coordinates (e) [9 marks] Reduce the equation to standard form (f) [3 marks] find its general solution.
02 Comsider the linca ecomd order partia i (a) [2 marks] determine the class of the equation (b) [2 marks]...
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....
dy (c) Write the first order differential equation x-y-x0 in standard form and find its general solution and then find the particular solution passing through (-2,4) (d) r (e) For what values of c do...
dy (c) Write the first order differential equation x-y-x0 in standard form and find its general solution and then find the particular solution passing through (-2,4) (d) r (e) For what values of c does the integadx converge? al-dx converge? x(In x) O Inrlde -1 In x lux 2e Ch) In In xdr
dy (c) Write the first order differential equation x-y-x0 in standard form and find its general solution and then find the particular solution passing through (-2,4)
(d)...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solutio...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
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....
Use the Fourier transform to find a solution of the ordinary differential equation u´´-u+2g(x) =0 where g∈L1. (The solu...
Use the Fourier transform to find a solution of the ordinary
differential equation u´´-u+2g(x) =0
where g∈L1. (The solution obtained this
way is the one that vanishes at ±∞. What is the general
solution?)
1. Use the Fourier transform to find a solution of the ordinary differential equation u" - u + 2g(x) = 0 where g E L. (The solution obtained this way is the one that vanishes at £oo. What is the general solution?) eg(y)dy eg(y)dy e Answer:...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r (x - 2)y 1, (2) r> 0. (a) Use the substitution y(x) = u(x)/x to show that the associated homogeneous equation ry" 2(r (x - 2)y 0 transforms into a linear constant-coefficient ODE for u(r) (b) Solve the linear constant-coefficient ODE obtained in Part (a) for u(x). Hence show that yeand y2= are solutions of the associated homogeneous ODE of equation (2). (c) Use...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
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]...
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 (x, y) plane (c) (d) [2 Marks] determine the characteristic coordinates (e) [9 marks] Reduce the equation to standard form (f) [3 marks] find its general solution.
02 Comsider the linca ecomd order partia i (a) [2 marks] determine the class of the equation (b) [2 marks]...
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....
dy (c) Write the first order differential equation x-y-x0 in standard form and find its general solution and then find the particular solution passing through (-2,4) (d) r (e) For what values of c does the integadx converge? al-dx converge? x(In x) O Inrlde -1 In x lux 2e Ch) In In xdr
dy (c) Write the first order differential equation x-y-x0 in standard form and find its general solution and then find the particular solution passing through (-2,4)
(d)...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
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....
Use the Fourier transform to find a solution of the ordinary
differential equation u´´-u+2g(x) =0
where g∈L1. (The solution obtained this
way is the one that vanishes at ±∞. What is the general
solution?)
1. Use the Fourier transform to find a solution of the ordinary differential equation u" - u + 2g(x) = 0 where g E L. (The solution obtained this way is the one that vanishes at £oo. What is the general solution?) eg(y)dy eg(y)dy e Answer:...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r (x - 2)y 1, (2) r> 0. (a) Use the substitution y(x) = u(x)/x to show that the associated homogeneous equation ry" 2(r (x - 2)y 0 transforms into a linear constant-coefficient ODE for u(r) (b) Solve the linear constant-coefficient ODE obtained in Part (a) for u(x). Hence show that yeand y2= are solutions of the associated homogeneous ODE of equation (2). (c) Use...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
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