4.1 First-Order ODEs and Separable Equations

First-order ODEs are equations involving the first derivative of a function. They come in various forms, including linear, nonlinear, and autonomous. Understanding these types helps in choosing the right solving method, like separation of variables.

These equations model real-world phenomena like population growth and radioactive decay. By applying initial conditions, we can find particular solutions that describe specific scenarios. This connects math to practical applications in science and engineering.

First-Order Ordinary Differential Equations (ODEs)

Classification of first-order ODEs

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• First-order ODEs involve the first derivative of the dependent variable with respect to the independent variable
  • General form: $\frac{dy}{dx} = f(x, y)$ where $y$ is the dependent variable and $x$ is the independent variable
• Linear first-order ODEs have the form: $\frac{dy}{dx} + P(x)y = Q(x)$
  • $P(x)$ and $Q(x)$ are functions of the independent variable $x$ only, not involving the dependent variable $y$
  • Example: $\frac{dy}{dx} + 3x^2y = \sin(x)$ is a linear first-order ODE
• Nonlinear first-order ODEs cannot be written in the linear form due to the presence of nonlinear terms involving $y$ or its derivatives
  • Example: $\frac{dy}{dx} = y^2 + x$ is a nonlinear first-order ODE because of the $y^2$ term
• Autonomous first-order ODEs have the form: $\frac{dy}{dx} = f(y)$ where the right-hand side depends only on the dependent variable $y$
  • Example: $\frac{dy}{dx} = y(1-y)$ is an autonomous first-order ODE (logistic equation)

Separation of variables technique

• Separable first-order ODEs have the form: $\frac{dy}{dx} = g(x)h(y)$ where $g(x)$ is a function of the independent variable $x$ and $h(y)$ is a function of the dependent variable $y$
  • Example: $\frac{dy}{dx} = x^2y^3$ is a separable first-order ODE with $g(x) = x^2$ and $h(y) = y^3$
• Separation of variables technique involves the following steps:
  1. Rearrange the equation to separate variables: $\frac{dy}{h(y)} = g(x)dx$
  2. Integrate both sides: $\int \frac{dy}{h(y)} = \int g(x)dx$
  3. Solve for $y$ as a function of $x$ by evaluating the integrals and applying algebraic manipulations
• Example: $\frac{dy}{dx} = x^2y^3$
  • Rearrange: $\frac{dy}{y^3} = x^2dx$
  • Integrate: $\int \frac{dy}{y^3} = \int x^2dx$
  • Solve: $-\frac{1}{2y^2} = \frac{x^3}{3} + C$, where $C$ is an arbitrary constant

Particular solutions with initial conditions

• General solution of a first-order ODE contains an arbitrary constant $C$ that represents a family of solutions
• Initial condition specifies the value of the dependent variable at a specific point, such as $y(x_0) = y_0$, where $x_0$ and $y_0$ are known values
  • Example: $y(0) = 1$ means the value of $y$ is 1 when $x = 0$
• Substitute the initial condition into the general solution to determine the value of $C$
• Replace $C$ in the general solution with the determined value to obtain the particular solution that satisfies the initial condition
• Example: Given the general solution $y = Ce^{2x}$ and the initial condition $y(0) = 3$, find the particular solution
  • Substitute: $3 = Ce^{2(0)} = C$
  • Replace: $y = 3e^{2x}$ is the particular solution

Applications of First-Order ODEs

Exponential growth and decay modeling

• Exponential growth and decay problems can be modeled using first-order ODEs of the form: $\frac{dy}{dt} = ky$, where $y$ is the quantity undergoing growth or decay and $k$ is the growth or decay constant
  • $k > 0$ for exponential growth (population growth, compound interest)
  • $k < 0$ for exponential decay (radioactive decay, cooling of objects)
• Solution to the exponential growth/decay ODE: $y(t) = y_0e^{kt}$, where $y_0$ is the initial value at $t = 0$
• Half-life ($t_{1/2}$) is the time required for the quantity to reduce to half its initial value in exponential decay
  • For exponential decay: $t_{1/2} = \frac{\ln(2)}{|k|}$, where $|k|$ is the absolute value of the decay constant
  • Example: If a radioactive substance decays with a half-life of 10 days, its decay constant is $k = -\frac{\ln(2)}{10}$
• Doubling time ($t_d$) is the time required for the quantity to double its initial value in exponential growth
  • For exponential growth: $t_d = \frac{\ln(2)}{k}$, where $k$ is the growth constant
  • Example: If a population grows with a doubling time of 5 years, its growth constant is $k = \frac{\ln(2)}{5}$