Differential Equations with SageMath in CoCalc - Part 2

# Essential SageMath setup for differential equations
from sage.all import *

# Configure plotting for better visualization
import matplotlib.pyplot as plt
%matplotlib inline

print(" SageMath environment loaded successfully!")
print(f" Plotting backend: {plt.get_backend()}")
print(" Ready to explore differential equations!")

# Define symbolic variables for population model
t = var('t')  # time variable
P = function('P')(t)  # population as a function of time
k = var('k')  # growth rate constant

# The differential equation: dP/dt = k*P
population_eq = diff(P, t) == k * P

print("Population Growth Differential Equation:")
print(f"   {population_eq}")
print()
print("Interpretation:")
print("   • Left side: Rate of population change")
print("   • Right side: Growth constant × Current population")
print("   • Meaning: Larger populations grow faster (exponential growth)")

population_solution = desolve(population_eq, P, ivar=t)

print("Solving the differential equation:")
print(f"   General solution: P(t) = {population_solution}")
print()
print("Understanding the solution:")
print("   • C₁ represents the initial population P(0)")
print("   • e^(kt) creates exponential growth/decay")
print("   • If k > 0: population grows exponentially")
print("   • If k < 0: population declines exponentially")
print("   • If k = 0: population remains constant")

# Visualize population growth for different scenarios
t_vals = srange(0, 5, 0.1)

# Create plots for different growth rates
p = Graphics()
colors = ['blue', 'red', 'green', 'purple']
scenarios = [0.5, 0.2, -0.1, -0.3]
labels = ['Fast growth (k=0.5)', 'Moderate growth (k=0.2)', 
         'Slow decline (k=-0.1)', 'Fast decline (k=-0.3)']

for i, k_val in enumerate(scenarios):
    P_vals = [1 * exp(k_val * t_val) for t_val in t_vals]
    p += list_plot(list(zip(t_vals, P_vals)), plotjoined=True,
                   color=colors[i], legend_label=labels[i], thickness=2)

p.axes_labels(['Time t', 'Population P(t)'])
p.show(title='Population Dynamics: $P(t)=P_0 e^{k t}$', legend_loc='upper left')

print("Key Insight: The same mathematical form describes both growth and decay!")

# Define variables for Newton's Law of Cooling
T = function('T')(t)  # temperature as function of time
T_room = var('T_room')  # room temperature (constant)
k_cooling = var('k_cooling')  # cooling constant

# Newton's Law of Cooling differential equation
cooling_eq = diff(T, t) == -k_cooling * (T - T_room)

print("Newton's Law of Cooling:")
print(f"   {cooling_eq}")
print()
print("Physical Interpretation:")
print("   • Negative sign: Temperature decreases when T > T_room")
print("   • Proportional to difference: Larger difference → faster cooling")
print("   • As T → T_room, cooling rate → 0 (approaches equilibrium)")

t = var('t')
k, T_room, T0 = var('k T_room T0')
# assume(T0, 'real')  # optional

T = function('T')(t)

# Newton's cooling: dT/dt = -k*(T - T_room)
cooling_eq = diff(T, t) == -k*(T - T_room)

# General solution
cooling_solution = desolve(cooling_eq, T, ivar=t)
print("Solving the cooling equation:")
print(f"   General solution: T(t) = {cooling_solution}")

# With initial condition T(0) = T0
cooling_solution_ic = desolve(cooling_eq, T, ics=[0, T0], ivar=t)
print(f"   With T(0) = T0: T(t) = {cooling_solution_ic}")

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## **From First-Order Examples to Classification and Solution Methods**

You've completed **First-Order Examples**! The population growth and cooling models demonstrate how differential equations capture the mathematics of change in real-world systems.

**What awaits in Part 3?**

Part 3 explores **Classification and Solution Methods** - learning to categorize different types of differential equations and the systematic approaches to solve them.

### **Continue Your Journey**

The concrete examples you've worked with will now be generalized into broader solution techniques and classification schemes.

**[Continue to Part 3: Classification and Solution Methods →](https://cocalc.com/share/public_paths/c0b7bff7fcdd845564cc6c77dd23db6736b99c77)**

**[Return to Main Course](https://cocalc.com/share/public_paths/43b79cbb7789fef95ac7cf057d8a27db19a84f34)**