Primer on Ordinary Differential Equations

Table of Contents

What is an ODE?

$$F \left( t, y, y'(t), y''(t), \ldots, y^{(n)}(t) \right) = 0$$

Where:

• $t$ is the independent variable (often time)
• $y$ is the dependent variable (the function we want to solve for)
• $y'(t)$ is the first derivative of $y$ with respect to $t$
• $y''(t)$ is the second derivative of $y$ with respect to $t$
• $y(t)^{(n)}$ is the $n$-th derivative of $y$ with respect to $t$
• $F$ is a function that relates these variables

The order is the highest derivative in the ODE. The degree is the exponent of the highest-order derivative (after clearing fractions/roots, only defined for polynomial ODEs). It is linear if the function and its derivatives appear to power 1, with no products. It is homogeneous if RHS = 0; otherwise non-homogeneous. The initial conditions are function/derivative values at a point for a unique solution.

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Methods of Solving Analytically

Separation of Variables

For first-order ODEs of the form:

$$\frac{dy}{dt} = g(t)h(y)$$

Rearranging gives:

$$\frac{1}{h(y)} dy = g(t) dt$$

Integrating both sides:

$$\int \frac{1}{h(y)} dy = \int g(t) dt$$

Example:

$$\frac{dy}{dt} = ky$$ $$\frac{1}{y} dy = k dt$$ $$\ln |y| = kt + C$$ $$y = Ce^{kt}$$

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Integrating Factors

For first-order linear ODEs of the form:

$$\frac{dy}{dt} + P(t)y = Q(t)$$

Multiply by the integrating factor:

$$\mu(t) = e^{\int P(t) dt}$$

Then:

$$\frac{d}{dt} \left( \mu(t)y \right) = \mu(t)Q(t)$$

Integrate both sides:

$$\int \frac{d}{dt} \left( \mu(t)y \right) dt = \int \mu(t)Q(t) dt$$ $$\mu(t)y = \int \mu(t)Q(t) dt + C$$ $$y = \frac{1}{\mu(t)} \left( \int \mu(t)Q(t) dt + C \right)$$

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Characteristic Equation

For linear ODEs with constant coefficients, such as:

$$a_n y^{(n)} + a_{n-1} y^{(n-1)} + \ldots + a_1 y' + a_0 y = 0$$

Assume a solution of the form:

$$y(t) = e^{rt}$$

Substituting into the ODE gives the characteristic equation:

$$a_n r^n + a_{n-1} r^{n-1} + \ldots + a_1 r + a_0 = 0$$

Solve for $r$ to find the roots. The general solution depends on the nature of the roots:

• Distinct Real Roots:

$$y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} + \ldots + C_n e^{r_n t}$$

• Repeated Roots:

$$y(t) = (C_1 + C_2 t + \ldots + C_n t^{n-1}) e^{r t}$$

• Complex Roots:

$$y(t) = e^{\alpha t} \left( C_1 \cos(\beta t) + C_2 \sin(\beta t) \right)$$

Where $r = \alpha + i\beta$ are the complex roots.

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Methods of Solving Numerically

Euler's Method

For first-order ODEs of the form:

$$\frac{dy}{dt} = f(t, y)$$

1. Choose a step size $h$ and initial condition $y(t_0) = y_0$.
2. Compute the next value:

$$y(t_0 + h) = y_0 + h f(t_0, y_0)$$

3. Repeat for subsequent points:

$$y(t_{n+1}) = y_n + h f(t_n, y_n)$$

4. Continue until the desired range is covered.

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Workflow

First Order ODEs

• Check if separable:
  • If yes → separate and integrate.
  • If no → check if linear:
    • If yes → use integrating factors.
    • If no → check if exact:
      • If yes → solve as exact.
      • If no → check if homogeneous:
        • If yes → try substitution.
        • If no → use numerical methods.

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