First-Order DEs (Separable & Homogeneous)

A Differential Equation (DE) is an equation containing an unknown function and one or more of its derivatives. Differential equations are the language of nature, describing how quantities change relative to one another.

Differential Equation

An equation involving an independent variable xx, a dependent variable yy, and one or more derivatives of yy with respect to xx.

Classifying Differential Equations

Before solving a DE, it is crucial to classify it to determine the appropriate solution method. The first major distinction is between Ordinary and Partial Differential Equations.

ODE vs. PDE

  • Ordinary Differential Equation (ODE): Involves derivatives with respect to only one independent variable. Example: dydx+5y=ex\frac{dy}{dx} + 5y = e^x
  • Partial Differential Equation (PDE): Involves partial derivatives with respect to two or more independent variables. Example: ∂2u∂x2+∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0

Classification Criteria for ODEs

  • Order: The order of a DE is the order of the highest derivative present in the equation.
  • Degree: The degree of a DE is the power of the highest derivative, provided the equation is a polynomial in its derivatives.
  • Linearity: A DE is linear if the dependent variable yy and all its derivatives appear to the first power and are not multiplied together or acted upon by non-linear functions (like sine, exp).
Examples:
  • dydx+5y=ex\frac{dy}{dx} + 5y = e^x (First-order, First-degree, Linear)
  • d2ydx2+(dydx)2+y=0\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + y = 0 (Second-order, First-degree, Non-linear)

Initial Value Problems (IVPs)

A differential equation typically has infinitely many solutions, forming a general solution with arbitrary constants (like +C+ C). To find a particular solution, we need additional information.

Initial Value Problem (IVP)

An IVP consists of a differential equation coupled with one or more initial conditions evaluated at the same point.
For a first-order DE: dydx=f(x,y)\frac{dy}{dx} = f(x,y) subject to y(x0)=y0y(x_0) = y_0.

Visualizing Solutions: Direction Fields

For a first-order DE y′=f(x,y)y' = f(x,y), we can visualize the solution without solving it analytically by plotting the slope f(x,y)f(x,y) at various points (x,y)(x,y). This creates a Slope Field (or Direction Field).

Slope Field Generator

Click anywhere on the graph to visualize a solution curve (trajectory) passing through that point.

Variable Separable Differential Equations

The simplest type of first-order DE is one where the variables can be separated on opposite sides of the equation.

Separable Equation

A first-order DE is separable if it can be factored into the form:
N(y) dy=M(x) dxN(y) \, dy = M(x) \, dx

Method of Separation of Variables

  1. Separate: Rearrange terms so that all yy's are with dydy and all xx's are with dxdx.
  2. Integrate: Integrate both sides: ∫N(y) dy=∫M(x) dx\int N(y) \, dy = \int M(x) \, dx.
  3. Solve: Add an arbitrary constant CC to one side and solve for yy explicitly if possible. If an IVP is given, plug in the initial conditions to solve for CC.

Homogeneous Differential Equations

A first-order DE is homogeneous if it can be written in the form M(x,y)dx+N(x,y)dy=0M(x,y)dx + N(x,y)dy = 0 where MM and NN are homogeneous functions of the same degree.

Homogeneous Function Test

A function f(x,y)f(x,y) is homogeneous of degree nn if for any parameter λ\lambda:
f(λx,λy)=λnf(x,y)f(\lambda x, \lambda y) = \lambda^n f(x,y)
Tip: If all terms in MM and NN have the same total degree of variables, the DE is homogeneous.

Solving Homogeneous DEs

  1. Substitute: Let y=vxy = vx (implies dy=vdx+xdvdy = vdx + xdv) OR x=vyx = vy (implies dx=vdy+ydvdx = vdy + ydv).
    • Use y=vxy = vx if NN is simpler.
    • Use x=vyx = vy if MM is simpler.
  2. Simplify: Substitute these into the original DE. The new equation in terms of vv and xx (or yy) will be separable.
  3. Integrate: Separate variables and integrate.
  4. Back-Substitute: Replace vv with y/xy/x (or x/yx/y) to get the final general solution.
Key Takeaways
  • A Differential Equation relates a function to its derivatives. ODEs have one independent variable; PDEs have multiple.
  • An Initial Value Problem (IVP) provides conditions to find a specific particular solution from the general solution.
  • Separation of Variables is the primary method for solving first-order DEs where xx and yy terms can be isolated.
  • Homogeneous DEs can be transformed into separable equations using the substitution y=vxy=vx or x=vyx=vy.
  • Slope Fields provide a geometric interpretation of solutions without solving analytically.
  • Always check the order and degree to classify the equation correctly before starting.
PreviousDerivatives of Parametric and Polar Curves - Examples & Applications
Quiz Me
NextFirst-Order DEs (Separable & Homogeneous) - Examples & Applications