```python import matplotlib.pyplot as plt from scipy.integrate import odeint from numpy import linalg as LA import numpy as np def model(X, t, m, g): y, v = X dydt = v dvdt = (m*g - 5*v - 120*y) / m return [dydt, dvdt] t0=0; tEnd=100 t=np.linspace(t0, tEnd, 50*(tEnd-t0)) m = 85 # mass y0 = 100.0 # initial height v0 = 0.0 # initial velocity g = 9.8 # acceleration due to gravity # solve ODE X0 = [y0, v0] X=odeint(model, X0, t, args=(m,g)) # args used for changing parameters without needing to change the whole code y = X[:, 0] v = X[:, 1] ``` ### Explanation This Python code is designed to solve a simple differential equation involving motion under gravity with damping effects. Here's a breakdown of the components: - **Imports:** - `matplotlib.pyplot` as `plt`: Used for plotting graphs (though not used directly in this snippet). - `odeint` from `scipy.integrate`: A function for integrating ordinary differential equations (ODEs). - `linalg` from `numpy` as `LA`: Provides linear algebra functions (not used in this snippet). - `numpy` as `np`: Used for numerical operations. - **Function Definition:** - `model(X, t, m, g)`: This function defines the differential equations for the system. It takes in: - `X`: A list containing the position `y` and velocity `v`. - `t`: Time variable. - `m`: Mass of the object. - `g`: Acceleration due to gravity. - Outputs the derivatives: position derivative `dydt` and velocity derivative `dvdt`. - **Parameters:** - `t0`, `tEnd`: Start and end times for the simulation. - `t`: Time points at which the solution is calculated, using `np.linspace`. - `m`: Mass of the object (85 units). - `y0`: Initial height (
```python import matplotlib.pyplot as plt from scipy.integrate import odeint from numpy import linalg as LA import numpy as np def model(X, t, m, g): y, v = X dydt = v dvdt = (m*g - 5*v - 120*y) / m return [dydt, dvdt] t0=0; tEnd=100 t=np.linspace(t0, tEnd, 50*(tEnd-t0)) m = 85 # mass y0 = 100.0 # initial height v0 = 0.0 # initial velocity g = 9.8 # acceleration due to gravity # solve ODE X0 = [y0, v0] X=odeint(model, X0, t, args=(m,g)) # args used for changing parameters without needing to change the whole code y = X[:, 0] v = X[:, 1] ``` ### Explanation This Python code is designed to solve a simple differential equation involving motion under gravity with damping effects. Here's a breakdown of the components: - **Imports:** - `matplotlib.pyplot` as `plt`: Used for plotting graphs (though not used directly in this snippet). - `odeint` from `scipy.integrate`: A function for integrating ordinary differential equations (ODEs). - `linalg` from `numpy` as `LA`: Provides linear algebra functions (not used in this snippet). - `numpy` as `np`: Used for numerical operations. - **Function Definition:** - `model(X, t, m, g)`: This function defines the differential equations for the system. It takes in: - `X`: A list containing the position `y` and velocity `v`. - `t`: Time variable. - `m`: Mass of the object. - `g`: Acceleration due to gravity. - Outputs the derivatives: position derivative `dydt` and velocity derivative `dvdt`. - **Parameters:** - `t0`, `tEnd`: Start and end times for the simulation. - `t`: Time points at which the solution is calculated, using `np.linspace`. - `m`: Mass of the object (85 units). - `y0`: Initial height (
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Below is a python code that allows us to solve a linear second-order non-homogeneous differential equation. We use the args command to allow us to change the parameter, m, without needing to rewrite the whole code. This results in a solution plot for the system corresponding to that particular value of m. But how would I generate a graph that allows me to plot multiple solutions at the same time? Write a code that will generate a plot allowing for multiple solutions for different values of m to be contained on the same graph.
![```python
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from numpy import linalg as LA
import numpy as np
def model(X, t, m, g):
y, v = X
dydt = v
dvdt = (m*g - 5*v - 120*y) / m
return [dydt, dvdt]
t0=0; tEnd=100
t=np.linspace(t0, tEnd, 50*(tEnd-t0))
m = 85 # mass
y0 = 100.0 # initial height
v0 = 0.0 # initial velocity
g = 9.8 # acceleration due to gravity
# solve ODE
X0 = [y0, v0]
X=odeint(model, X0, t, args=(m,g)) # args used for changing parameters without needing to change the whole code
y = X[:, 0]
v = X[:, 1]
```
### Explanation
This Python code is designed to solve a simple differential equation involving motion under gravity with damping effects. Here's a breakdown of the components:
- **Imports:**
- `matplotlib.pyplot` as `plt`: Used for plotting graphs (though not used directly in this snippet).
- `odeint` from `scipy.integrate`: A function for integrating ordinary differential equations (ODEs).
- `linalg` from `numpy` as `LA`: Provides linear algebra functions (not used in this snippet).
- `numpy` as `np`: Used for numerical operations.
- **Function Definition:**
- `model(X, t, m, g)`: This function defines the differential equations for the system. It takes in:
- `X`: A list containing the position `y` and velocity `v`.
- `t`: Time variable.
- `m`: Mass of the object.
- `g`: Acceleration due to gravity.
- Outputs the derivatives: position derivative `dydt` and velocity derivative `dvdt`.
- **Parameters:**
- `t0`, `tEnd`: Start and end times for the simulation.
- `t`: Time points at which the solution is calculated, using `np.linspace`.
- `m`: Mass of the object (85 units).
- `y0`: Initial height (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d0ee2cb-cfe2-4eb9-a097-d38324436758%2F0e2d4bea-1891-442b-9e6c-9c8d33a435bb%2F14a8fh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:```python
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from numpy import linalg as LA
import numpy as np
def model(X, t, m, g):
y, v = X
dydt = v
dvdt = (m*g - 5*v - 120*y) / m
return [dydt, dvdt]
t0=0; tEnd=100
t=np.linspace(t0, tEnd, 50*(tEnd-t0))
m = 85 # mass
y0 = 100.0 # initial height
v0 = 0.0 # initial velocity
g = 9.8 # acceleration due to gravity
# solve ODE
X0 = [y0, v0]
X=odeint(model, X0, t, args=(m,g)) # args used for changing parameters without needing to change the whole code
y = X[:, 0]
v = X[:, 1]
```
### Explanation
This Python code is designed to solve a simple differential equation involving motion under gravity with damping effects. Here's a breakdown of the components:
- **Imports:**
- `matplotlib.pyplot` as `plt`: Used for plotting graphs (though not used directly in this snippet).
- `odeint` from `scipy.integrate`: A function for integrating ordinary differential equations (ODEs).
- `linalg` from `numpy` as `LA`: Provides linear algebra functions (not used in this snippet).
- `numpy` as `np`: Used for numerical operations.
- **Function Definition:**
- `model(X, t, m, g)`: This function defines the differential equations for the system. It takes in:
- `X`: A list containing the position `y` and velocity `v`.
- `t`: Time variable.
- `m`: Mass of the object.
- `g`: Acceleration due to gravity.
- Outputs the derivatives: position derivative `dydt` and velocity derivative `dvdt`.
- **Parameters:**
- `t0`, `tEnd`: Start and end times for the simulation.
- `t`: Time points at which the solution is calculated, using `np.linspace`.
- `m`: Mass of the object (85 units).
- `y0`: Initial height (
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