First Order Differential Equations

• The text provides information about first order differential equations

Differential Equations: Definitions and Classifications

• A differential equation relates an unknown function to its derivatives.
• Example: $\frac{dy}{dx} = 5x^2 + 3$.

Classification of Differential Equations

• Type:
  • Ordinary Differential Equations (ODE): The unknown function depends on one variable.
    • Example: $\frac{dy}{dx} + y = x$.
  • Partial Differential Equations (PDE): The unknown function depends on multiple variables.
    • Example: $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$.
• Order: The order is the highest derivative in the equation.
  • Examples:
    • Order 1: $\frac{dy}{dx} + y = x$.
    • Order 2: $\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$.
• Linearity: The function and its derivatives appear linearly.
  • Examples:
    • Linear: $\frac{dy}{dx} + y = x$.
    • Non-linear: $\frac{dy}{dx} + y^2 = x$.

Solution of a Differential Equation

• A solution is a function that satisfies the equation when substituted.
• Example: $y = e^{-x}$ solves $\frac{dy}{dx} + y = 0$, because $\frac{d}{dx}(e^{-x}) + e^{-x} = -e^{-x} + e^{-x} = 0$.

Types of Solutions

• General Solution: Contains arbitrary constants.
• Particular Solution: Obtained by assigning specific values to constants in the general solution.
• Singular Solution: Cannot be obtained from the general solution.

First-Order Differential Equations

• Can be written as: $\frac{dy}{dx} = f(x, y)$

Separable Equations

• Can be written as: $g(y) \frac{dy}{dx} = f(x)$
• Solve by integrating both sides with respect to $x$: $\int g(y) dy = \int f(x) dx$.
• Example:
  • $\frac{dy}{dx} = xy$.
  • $\frac{dy}{y} = x dx$.
  • $ln|y| = \frac{x^2}{2} + C$.
  • $y = Ke^{\frac{x^2}{2}}$.

Homogeneous Equations

• Can be written as: $\frac{dy}{dx} = F(\frac{y}{x})$.
• Use substitution $v = \frac{y}{x}$, so $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$.
• Results in a separable equation with $v$ and $x$.
• Example:
  • $\frac{dy}{dx} = \frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x}$.
  • Substitute $v = \frac{y}{x}$: $v + x \frac{dv}{dx} = \frac{1}{v} + v$.
  • $x \frac{dv}{dx} = \frac{1}{v}$.
  • $v dv = \frac{1}{x} dx$.
  • $\frac{v^2}{2} = ln|x| + C$.
  • $\frac{y^2}{2x^2} = ln|x| + C$.
  • $y^2 = 2x^2(ln|x| + C)$.

Exact Equations

• Can be written as: $M(x, y) dx + N(x, y) dy = 0$, where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
• Find $F(x, y)$ such that $\frac{\partial F}{\partial x} = M(x, y)$ and $\frac{\partial F}{\partial y} = N(x, y)$.
• Solution: $F(x, y) = C$.

Applications of Differential Equations

• Population Growth and Decay
  • Model: $\frac{dP}{dt} = kP$.
    • $P(t)$: population at time $t$.
    • $k$: growth rate ($k > 0$) or decay rate ($k < 0$).
  • Solution: $P(t) = P_0e^{kt}$, with $P_0$ being the initial population.
• Newton's Law of Cooling
  • Model: $\frac{dT}{dt} = k(T - T_a)$.
    • $T(t)$: object's temperature at time $t$.
    • $T_a$: ambient temperature.
    • $k$: proportionality constant (negative).
  • Solution: $T(t) = T_a + (T_0 - T_a)e^{kt}$, with $T_0$ as the initial temperature.
• Electrical Circuits
  • Used to model circuits.
  • Example: RL circuit in series: $L \frac{di}{dt} + Ri = E(t)$.
    • $L$: inductance.
    • $R$: resistance.
    • $E(t)$: electromotive force (voltage).
  • Solution depends on the form of $E(t)$.