Cosider the second order initial value problem y y' exp(-3y) - 2 ty+t, te [2, 4],...

 

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Cosider the second order initial value problem y" y' exp(-3y) - 2 ty+t², te [2, 4], y(2)=3, y' (2) = 1. (a) [1 mark] To convert this into a system of first order ODEs, we introduce ₁ = y and r₂ =y'. Then we obtain the first order system a x' = f(t, x) = [f(t,z)]. where fi(t, x) = 1₂ f₂(t, x) = Your expressions should be in terms of t, ₁ and ₂ and should not contain y. Use square brackets to enter subscripts, e.g., x[2] for #₂. Remember to type the multiplication symbol * whenever appropriate. Click on the preview button to double check your answer. (b) [1 mark] The initial condition for the first order system æ' = f(t, x) is 20= Use Maple syntax to enter a vector, e.g., <1,2> for [2] (c) [1 mark] Euler's method for this problem can be formulated as *n+1 = n + hf (tn, n), or in terms of the components, In+1,1 = n1 +han,2 In+1,2 = £n2 + h([ Use square brackets to enter subscripts, e.g., t[n] for t, and x[n,1] for 1 Remember to type the multiplication symbol whenever appropriate. Click on the preview button to double check your answer. (d) [4 marks] Define a MATLAB anonymous function for f. Then use the MATLAB function rk4.m from the lectures to solve this initial value problem with step size h = 0.05. Complete the following entries. Give your answers to 4 significant figures. y(2.15) Number y(2.35) Number (e) [3 marks] Replace rk4.m by the MATLAB function ode45 with the default tolerance setting. Complete the following entries. Give your answers to 4 significant figures. Number y(4) y' (4) Number Cosider the second order initial value problem y" y' exp(-3y) - 2 ty+t², te [2, 4], y(2)=3, y' (2) = 1. (a) [1 mark] To convert this into a system of first order ODEs, we introduce ₁ = y and r₂ =y'. Then we obtain the first order system a x' = f(t, x) = [f(t,z)]. where fi(t, x) = 1₂ f₂(t, x) = Your expressions should be in terms of t, ₁ and ₂ and should not contain y. Use square brackets to enter subscripts, e.g., x[2] for #₂. Remember to type the multiplication symbol * whenever appropriate. Click on the preview button to double check your answer. (b) [1 mark] The initial condition for the first order system æ' = f(t, x) is 20= Use Maple syntax to enter a vector, e.g., <1,2> for [2] (c) [1 mark] Euler's method for this problem can be formulated as *n+1 = n + hf (tn, n), or in terms of the components, In+1,1 = n1 +han,2 In+1,2 = £n2 + h([ Use square brackets to enter subscripts, e.g., t[n] for t, and x[n,1] for 1 Remember to type the multiplication symbol whenever appropriate. Click on the preview button to double check your answer. (d) [4 marks] Define a MATLAB anonymous function for f. Then use the MATLAB function rk4.m from the lectures to solve this initial value problem with step size h = 0.05. Complete the following entries. Give your answers to 4 significant figures. y(2.15) Number y(2.35) Number (e) [3 marks] Replace rk4.m by the MATLAB function ode45 with the default tolerance setting. Complete the following entries. Give your answers to 4 significant figures. Number y(4) y' (4) Number

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a Given 2nd order IVP as d2ydt2 exp3y dydt 2ty t2 If x1 y x2 dydt we can say that dx2 dt ex... View the full answer