11(0.1) = 0, 14(r,t) = 0, 1 〉 0 t 〉 0 (a) 14-14x = 0, 0〈x〈L, t > 0 a(0.1) 0, 14(T.t)+γ.11(T,t)-O, t > 0 t>O, γǐs constant.
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2. Find non-trivial (non-zero) product solutions for each of the following homogeneous bound ary ...
Thank you. 1. Find all solutions of each of the following homogeneous boundary value problems. (a)...
Thank you.
1. Find all solutions of each of the following homogeneous boundary value problems. (a) y"+y=0, y(0) = 0, y'(T) = 0. (b) y" – 2y' + 2y = 0, y(0) = 0, y(T) = 0. (c) y + y = 0, y(0) = 0, y(L) = 0, where L is a positive real number.
2. In this question you will find the non-zero separable solutions elar,t-M(r)N(G) of the Klein Gonlon equation 01...
2. In this question you will find the non-zero separable solutions elar,t-M(r)N(G) of the Klein Gonlon equation 01 -03 subject to the boundary conditions e(0, t) = ψ(r, t) = 0. 3 points)(a) Show that the problem is equivalent to finding the possible non-zero solutions of M(1-A)M( N"(t)-AN(t) where λ is the separation constant to be determined. (2 points) (b) Let Л -1. Show that if A-: 0 then M(z)-0 is the only solution. {c) Show that if Λ =-k,...
Find the general solutions to the following non-homogeneous Cauchy-Euler equation using variation of parameters. 22" +...
Find the general solutions to the following non-homogeneous Cauchy-Euler equation using variation of parameters. 22" + tz' + 362 = - tan (6 Int) z(t)= (Use parentheses to clearly denote the argument of each function.)
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Previous Problem Problem List Next Problem (1 point) Let's find the general solution to z2y"-5zy, + 8y-(2-P) using reduction o of order (1) First find a non-trivial solution to the comple...
Previous Problem Problem List Next Problem (1 point) Let's find the general solution to z2y"-5zy, + 8y-(2-P) using reduction o of order (1) First find a non-trivial solution to the complementary equation z' smaller power m. 5zy' +8y0 of the form z. There are two possibilities, pe (2) Now set u = tizm and determine a first order equation (in standard form) that ,' t' must satisfy (3) Solve this for z using cl as the arbitrary constant 4) Solve...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
5. Repeat the same questions in 4.) for the ODE Py"- tt+2)y+(t+2)y2t3, (t>0) (a) Find the general solution of the homogeneous ODE y"- 5y +6y 0. Particularly find yi and (b) Find the...
5. Repeat the same questions in 4.) for the ODE Py"- tt+2)y+(t+2)y2t3, (t>0) (a) Find the general solution of the homogeneous ODE y"- 5y +6y 0. Particularly find yi and (b) Find the equivalent nonhomogeneous system of first order with the chan of variable y (c) Show that (nvand 2( re solutions of the homogeneous system of ODEs (d) Find the variation of parameters equations that have to be satisfic 1 for y(t) vi(t)u(t) + (e) Find the variation of...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is a fundamental set of solutions for the homogeneous problem y"+p(r)y' +(xy-0. Then the formula for the particular solution using the method of variation of parameters is are where W(z) is the Wronskian given by the determinant where ufe) and u ,-1-nent), d dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. So we have- Wed and...
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0 and y(2) 0 For certain discrete values of a, this equation can have non-zero solutions. Find the three...
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0 and y(2) 0 For certain discrete values of a, this equation can have non-zero solutions. Find the three smallest values of a for which this is the case. Enter your answers in increasing order. a2 , аз Note: You can earn partial credit on this problem
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0...
2. Consider a thin rod of length L = π (so that 0 x-7) with a...
2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and...
Thank you.
1. Find all solutions of each of the following homogeneous boundary value problems. (a) y"+y=0, y(0) = 0, y'(T) = 0. (b) y" – 2y' + 2y = 0, y(0) = 0, y(T) = 0. (c) y + y = 0, y(0) = 0, y(L) = 0, where L is a positive real number.
2. In this question you will find the non-zero separable solutions elar,t-M(r)N(G) of the Klein Gonlon equation 01 -03 subject to the boundary conditions e(0, t) = ψ(r, t) = 0. 3 points)(a) Show that the problem is equivalent to finding the possible non-zero solutions of M(1-A)M( N"(t)-AN(t) where λ is the separation constant to be determined. (2 points) (b) Let Л -1. Show that if A-: 0 then M(z)-0 is the only solution. {c) Show that if Λ =-k,...
Find the general solutions to the following non-homogeneous Cauchy-Euler equation using variation of parameters. 22" + tz' + 362 = - tan (6 Int) z(t)= (Use parentheses to clearly denote the argument of each function.)
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Previous Problem Problem List Next Problem (1 point) Let's find the general solution to z2y"-5zy, + 8y-(2-P) using reduction o of order (1) First find a non-trivial solution to the complementary equation z' smaller power m. 5zy' +8y0 of the form z. There are two possibilities, pe (2) Now set u = tizm and determine a first order equation (in standard form) that ,' t' must satisfy (3) Solve this for z using cl as the arbitrary constant 4) Solve...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
5. Repeat the same questions in 4.) for the ODE Py"- tt+2)y+(t+2)y2t3, (t>0) (a) Find the general solution of the homogeneous ODE y"- 5y +6y 0. Particularly find yi and (b) Find the equivalent nonhomogeneous system of first order with the chan of variable y (c) Show that (nvand 2( re solutions of the homogeneous system of ODEs (d) Find the variation of parameters equations that have to be satisfic 1 for y(t) vi(t)u(t) + (e) Find the variation of...
(1 point) In general for a non-homogeneous problem " ()y r)y-f(x) assume that yi, ye is a fundamental set of solutions for the homogeneous problem y"+p(r)y' +(xy-0. Then the formula for the particular solution using the method of variation of parameters is are where W(z) is the Wronskian given by the determinant where ufe) and u ,-1-nent), d dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. So we have- Wed and...
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0 and y(2) 0 For certain discrete values of a, this equation can have non-zero solutions. Find the three smallest values of a for which this is the case. Enter your answers in increasing order. a2 , аз Note: You can earn partial credit on this problem
(1 point) Let a be a real constant. Consider the equation dx2 dx with boundary conditions y(0)0...
2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and...
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