Ordinary Differential Equations

1. Review of Ordinary Differential Equations (ODE)

Intuition

An Ordinary Differential Equation (ODE) relates a function $y(x)$ to its derivatives. The general form of an ODE is:

$$F(x,y,y^{\prime },y^{″},\dots ,y^{(n)})=0.$$

ODEs are classified based on:

• Order: The highest derivative present.
• Linearity: Whether the equation can be written as a linear combination of the dependent variable and its derivatives.

Example

$$y^{\prime }+y=0\text{(First-order linear ODE)}.$$

The solution is $y=C{e}^{-x}$, where $C$ is a constant.

Python Visualization

import numpy as np
import matplotlib.pyplot as plt

# Define the solution
def y(x, C):
    return C * np.exp(-x)

# Plot
x = np.linspace(0, 5, 500)
C = 1
plt.plot(x, y(x, C), label='y = Ce^{-x}')
plt.title('Solution of y' + y = 0')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.show()

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2. Linear Equations

Intuition

A first-order linear ODE has the form:

$$y^{\prime }+P(x)y=Q(x).$$

The solution is obtained using an integrating factor:

$$\mu (x)=e^{\int P(x)dx},y=\frac{1}{\mu (x)}\int \mu (x)Q(x)dx.$$

Example

Solve $y^{\prime }+y=x$:

$$P(x)=1,Q(x)=x,\mu (x)=e^x.$$

$$y=e^{-x}\int e^{x}xdx=e^{-x}(x{e}^x-e^x)+C=x-1+C{e}^{-x}.$$

Python Visualization

from sympy import symbols, Function, Eq, exp, dsolve

# Define symbols
x = symbols('x')
y = Function('y')

# Define ODE
otde = Eq(y(x).diff(x) + y(x), x)

# Solve ODE
solution = dsolve(otde)
print(f"Solution: {solution}")

# Plot
C = 1
x_vals = np.linspace(0, 5, 500)
y_vals = x_vals - 1 + C * np.exp(-x_vals)
plt.plot(x_vals, y_vals, label='y = x - 1 + Ce^{-x}')
plt.title('Solution of y' + y = x')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.show()

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3. Second-Order Linear Equations

Intuition

A second-order linear ODE has the form:

$$a{y}^{″}+b{y}^{\prime }+cy=0.$$

The solution depends on the roots of the characteristic equation $a{r}^2+br+c=0$:

1. Distinct real roots: $y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$.
2. Repeated root: $y=(C_1+C_{2}x)e^{r_{1}x}$.
3. Complex roots: $y=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$.

Example

Solve $y^{″}-3y^{\prime }+2y=0$:

$$\text{Characteristic equation: }{r}^2-3r+2=0,r=1,2.$$

$$y=C_1{e}^x+C_2{e}^{2x}.$$

Python Visualization

# Define ODE
otde2 = Eq(y(x).diff(x, 2) - 3*y(x).diff(x) + 2*y(x), 0)

# Solve ODE
solution2 = dsolve(otde2)
print(f"Solution: {solution2}")

# Plot
C1, C2 = 1, 1
x_vals = np.linspace(0, 5, 500)
y_vals = C1 * np.exp(x_vals) + C2 * np.exp(2 * x_vals)
plt.plot(x_vals, y_vals, label='y = C1 e^x + C2 e^{2x}')
plt.title('Solution of y'' - 3y' + 2y = 0')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.show()

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4. Non-Homogeneous Linear Equations

Intuition

A second-order non-homogeneous ODE has the form:

$$a{y}^{″}+b{y}^{\prime }+cy=G(x).$$

The solution is:

$$y=y_h+y_p$$

where:

• $y_h$: Solution of the homogeneous equation.
• $y_p$: Particular solution.

Example

Solve $y^{″}-y=e^x$:

$$y_h=C_1{e}^x+C_2{e}^{-x},y_p=\frac{1}{2}x{e}^x.$$

$$y=C_1{e}^x+C_2{e}^{-x}+\frac{1}{2}x{e}^x.$$

Python Visualization

# Define ODE
otde3 = Eq(y(x).diff(x, 2) - y(x), exp(x))

# Solve ODE
solution3 = dsolve(otde3)
print(f"Solution: {solution3}")

# Plot
C1, C2 = 1, 1
x_vals = np.linspace(0, 5, 500)
y_vals = C1 * np.exp(x_vals) + C2 * np.exp(-x_vals) + 0.5 * x_vals * np.exp(x_vals)
plt.plot(x_vals, y_vals, label='y = C1 e^x + C2 e^{-x} + 0.5x e^x')
plt.title('Solution of y'' - y = e^x')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.show()