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Given the differential equation: Assume that there exist a p(x,y) such that u(x,y) - f( p(x,y))...
Determine if the given function y- f(x) is a solution of the accompanying differential equation D...
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and u...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x,y) є R2 : 0 a, 0-y-b} with three boundary x conditions on Ф: obtain conditions on F and G on those boundaries where conditions on Ф are given. (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive in...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Partial differential equation-wave equation
One end of the pipe is closed, which corresponds to the boundary conditionu(0, t) = 0, for t > 0. The other end of the pipe is open, which correspondsto the boundary condition ux(L, t) = 0, for t > 0.(a) Suppose that µ < 0, so µ = −k^2 for some k > 0. Find the non-trivial solution X(x) that satisfies equations (3), stating clearly what values k is allowed to take.(b) Write down the general solution of equation...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0,...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
(1 point) A Bernoulli differential equation is one of the form dy dc + P(x)y= Q(x)y"...
(1 point) A Bernoulli differential equation is one of the form dy dc + P(x)y= Q(x)y" Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation du dr +(1 – n)P(x)u = (1 - nQ(x). Consider the initial value problem xy + y = 3xy’, y(1) = -8. (a) This differential equation can be written in the form (*)...
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:...
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and...
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and g are smooth (Co) functions. Let 7(t) be the solution of the initial value problem with γ(0) = (1,2). Prove or give a counterexample to the statement that the w-limit set of γ can contain more than one critical point.
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y)...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x,y) є R2 : 0 a, 0-y-b} with three boundary x conditions on Ф: obtain conditions on F and G on those boundaries where conditions on Ф are given. (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
(1 point) A Bernoulli differential equation is one of the form dy dc + P(x)y= Q(x)y" Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation du dr +(1 – n)P(x)u = (1 - nQ(x). Consider the initial value problem xy + y = 3xy’, y(1) = -8. (a) This differential equation can be written in the form (*)...
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:...
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and g are smooth (Co) functions. Let 7(t) be the solution of the initial value problem with γ(0) = (1,2). Prove or give a counterexample to the statement that the w-limit set of γ can contain more than one critical point.
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
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