Consider the ODE y + 2y' + 9y = 0 with the initial conditions y(0) =...

  

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Consider the ODE y" + 2y' + 9y = 0 with the initial conditions y(0) = 0 and y'(0) = 1. First, solve it analytically. Next rewite it as a coupled set of first-order ODE's. Now, write a computer code to integrate it using the backward Euler scheme, i.e. for each equation dA/dt = RHS, the scheme is written A, - An-1 = h RHS,, where h is the uniform step size. Write the resulting set of equations in matrix form and write a code to solve it. Finally, plot out the analytical and numerical solution over the range [0, 7). Try a few values of h and see what happens. Consider the ODE y" + 2y' + 9y = 0 with the initial conditions y(0) = 0 and y'(0) = 1. First, solve it analytically. Next rewite it as a coupled set of first-order ODE's. Now, write a computer code to integrate it using the backward Euler scheme, i.e. for each equation dA/dt = RHS, the scheme is written A, - An-1 = h RHS,, where h is the uniform step size. Write the resulting set of equations in matrix form and write a code to solve it. Finally, plot out the analytical and numerical solution over the range [0, 7). Try a few values of h and see what happens.