i need help please

Question

Newton's law of cooling.

dT/dt=−K(T−Ta)

Resolve this ODE using the scipy library.

Answer 1

Let me give you an example so you can figure it out easily

In [2]: from sympy import *

Example: Newton's Law of Cooling (160-degree turkey removed from the oven in a 70-degree room, the temperature is 155 degrees after 5 minutes. When will the temperature be 140 degrees?)

In [3]: # FIRST STEP: Solve the IVP from Newton's Law of Cooling: dT/dt = k*(T-70),

T(0)=160

k,t=symbols('k t')

T=Function('T')

deq=diff(T(t),t)-k*(T(t)-70)

Tsoln=dsolve(deq,T(t),ics={T(0):160})

print('Solution of the ODE is',Tsoln)

In [10]: # SECOND STEP: Use the 5 minute temperature to find k

keqn=Tsoln.rhs.subs(t,5)-155 # NOTE: 'rhs' refers to the right hand side o

f the equation

ksoln=solve(keqn,k)

print(ksoln) # last (5th) solution is real: remember Python starts counting

at ZERO!

print('Using the 5 minute temperature reading,',Tsoln.subs(k,ksoln[4]))

In [12]: # Finally, use the new equation to solve for t when T = 140

Teqn=Tsoln.rhs.subs(k,ksoln[4])

tcooled=solve(Teqn-140,t)

print(tcooled) #only one solution this time...the 0th one

print('The turkey is cool enough to eat after',tcooled[0],'or',tcooled[0].e

valf(),'minutes.')

Example 2: IVP (based on rate in - rate out) is

y' = 2a - 2y/100, y(0)=0

In [13]: matplotlib notebook

Solution of the ODE is Eq(T(t), 90*exp(k*t) + 70)

[log(17**(1/5)*2**(4/5)*3**(3/5)/6) - 4*I*pi/5, log(17**(1/5)*2**(4/5)*3**

(3/5)/6) - 2*I*pi/5, log(17**(1/5)*2**(4/5)*3**(3/5)/6) + 2*I*pi/5, log(17*

*(1/5)*2**(4/5)*3**(3/5)/6) + 4*I*pi/5, log(7344**(1/5)/6)]

Using the 5 minute temperature reading, Eq(T(t), 90*exp(t*log(7344**(1/5)/

6)) + 70)

[log((7/9)**(1/log(7344**(1/5)/6)))]

The turkey is cool enough to eat after log((7/9)**(1/log(7344**(1/5)/6))) o

r 21.9840274945891 minutes.