ode examples

main.md

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+# Excersices
+ Find a numerical solution to the following differential equations with the associated initial conditions. Expand the requested time horizon until the solution reaches a steady state. Show a plot of the states ($x(t)$ and/or $y(t)$). Report the final value of each state as $t \rightarrow \infty$.
+
+
+## Problem 1:
+
+$$\frac{dy(t)}{dt} = -y(t)+1 $$
+with, 
+$$ y(0)=0$$
+
+This equation can be solved using the separation of variables method:
+
+$$y(t) = 1-e^{-t}$$
+
+
+```python
+import numpy as np
+from scipy.integrate import odeint
+import matplotlib.pyplot as plt
+
+# function that returns dy/dt
+def model(y,t):
+    dydt = -y + 1.0
+    return dydt
+
+# initial condition
+y0 = 0
+
+# time points
+t = np.linspace(0,5)
+
+# solve ODE
+y = odeint(model,y0,t)
+ySol = 1-np.exp(-t)
+# plot results
+plt.plot(t,y,'ok', label='ODE')
+plt.plot(t,ySol, '.:r', label='ySol')
+plt.legend()
+plt.xlabel('time')
+plt.ylabel('y(t)')
+plt.show()
+```
+
+
+    
+![png](main_files/main_2_0.png)
+    
+
+
+# Problem 2
+$$5\frac{dy}{dt}=-y(t)+u(t)$$
+
+with, 
+$$y(0)=1$$
+
+and $u$ step at $t=2$
+
+
+```python
+# function that returns dy/dt
+def model(y,t):
+    # u steps from 0 to 2 at t=10
+    if t<10.0:
+        u = 0
+    else:
+        u = 2
+    dydt = (-y + u)/5.0
+    return dydt
+
+# initial condition
+y0 = 1
+
+# time points
+n = 40
+t = np.linspace(0,n-1,n)
+# solve ODE
+y = odeint(model,y0,t)
+
+# plot results
+plt.plot(t,y,'r-',label='Output (y(t))')
+plt.plot([0,10,10,40],[0,0,2,2],'b:',label='Input (u(t))')
+plt.ylabel('values')
+plt.xlabel('time')
+plt.legend(loc='best')
+plt.show()
+```
+
+
+    
+![png](main_files/main_4_0.png)
+    
+
+
+
+```python
+# Define the step function u(t)
+def u(t):
+    return 0 if t < 10.0 else 2  # Step from 0 to 2 at t = 10
+
+# function that returns dy/dt
+def model(y,t):
+    dydt = (-y + u(t))/5.0
+    return dydt
+
+# initial condition
+y0 = 1
+
+# time points
+n = 20
+t = np.linspace(0,n-1,n)
+
+# solve ODE
+y = odeint(model,y0,t)
+# Compute u(t) for all values in t
+u_values = np.array([u(ti) for ti in t])  # Evaluate u at each time point
+
+
+# plot results
+plt.plot(t,y,'.:r',label='Output (y(t))')
+plt.plot(t, u_values, 'g-', linewidth=2, label="u(t)")
+plt.axvline(x=10, color='r', linestyle='--', label="Step at t=10")
+plt.ylabel('values')
+plt.xlabel('time')
+plt.legend(loc='best')
+plt.show()
+
+```
+
+
+    
+![png](main_files/main_5_0.png)
+    
+
+
+
+```python
+
+```