Write a Python code for the following:

a) Evaluate the solution to the Initial Value problem x= x² ,x(1)= 1. Where time t = 1.5 and find which valve of h = 10^-k is needed to find x(1.5),With a precision of 10^-6

b) Show numerically that the local error of the Euler method is of order 2.
 

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This is the simplest numerical method to solve the IVP i= f(t, x), z(to) = 20. It is just a direct consequence of Taylor's series expansion formula: e(t + h) = x(t) + (t)h + (7) h ,x(t))h + (fi(7,x(7)) + f(7, x(T))ƒz(T,x(7)))h² for some to ST≤ to +h. -h² = The step of explicit Euler's method is precisely *n+1 = n + hf(tn, n) Ant=to th and it is clearly of local order 2. Below is an implementation of the Euler method used to evaluate numerically (1.5) for the solution of the ODE -cos(nt), 2(0)=√5/2. This particular implementation subdivides the "time" interval [0, 1.5] in 10 subintervals, namely h = 0.15. The code saves all intermediate steps and plots them together with the corresponding values of the exact solution z(t)=√1.25 + sin(xt). This is the simplest numerical method to solve the IVP i= f(t, x), z(to) = 20. It is just a direct consequence of Taylor's series expansion formula: e(t + h) = x(t) + (t)h + (7) h ,x(t))h + (fi(7,x(7)) + f(7, x(T))ƒz(T,x(7)))h² for some to ST≤ to +h. -h² = The step of explicit Euler's method is precisely *n+1 = n + hf(tn, n) Ant=to th and it is clearly of local order 2. Below is an implementation of the Euler method used to evaluate numerically (1.5) for the solution of the ODE -cos(nt), 2(0)=√5/2. This particular implementation subdivides the "time" interval [0, 1.5] in 10 subintervals, namely h = 0.15. The code saves all intermediate steps and plots them together with the corresponding values of the exact solution z(t)=√1.25 + sin(xt).

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