First-Order DEs (Separable & Homogeneous) - Examples & Applications

Conceptual Examples: Classification and Direction Fields

Classifying Ordinary vs. Partial Differential Equations

Determine whether the following differential equations are Ordinary Differential Equations (ODEs) or Partial Differential Equations (PDEs). Explain your reasoning.

a) md2xdt2+kx=0m\frac{d^2x}{dt^2} + kx = 0

b) ∂u∂t=α2(∂2u∂x2+∂2u∂y2)\frac{\partial u}{\partial t} = \alpha^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)

Classification Solution

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Determining Order, Degree, and Linearity

Classify the following ODE by determining its order, degree, and linearity.

(d3ydx3)2+4xdydx−y=ex\left(\frac{d^3y}{dx^3}\right)^2 + 4x \frac{dy}{dx} - y = e^x

Order, Degree, Linearity Solution

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Analyzing a Direction Field

Consider the differential equation dydx=x−y\frac{dy}{dx} = x - y. Without solving the equation, describe the behavior of the solution curve that passes through the initial point (0,0)(0, 0) by analyzing the direction field. What happens as xx becomes large?

Direction Field Analysis

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Variable Separable Differential Equations

Separable DE (Basic)

Find the general solution of the differential equation:

dydx=−xy\frac{dy}{dx} = -\frac{x}{y}

Separable DE Solution

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Separation of Variables (Intermediate)

Find the general solution of the differential equation:

dydx=1+y21+x2\frac{dy}{dx} = \frac{1+y^2}{1+x^2}

Separation of Variables Solution

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Separable DE (Advanced)

Find the general solution to the differential equation:

xe−ysin⁡(x) dx−y dy=0x e^{-y} \sin(x) \, dx - y \, dy = 0

Advanced Separable Solution

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Homogeneous Differential Equations

Homogeneous DE (Intermediate)

Solve the differential equation:

(x2+y2)dx−2xy dy=0(x^2 + y^2)dx - 2xy \, dy = 0

Homogeneous DE Solution

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Homogeneous DE (Advanced)

Solve the differential equation:

xdydx=y+x2−y2for x>0x \frac{dy}{dx} = y + \sqrt{x^2 - y^2} \quad \text{for } x > 0

Advanced Homogeneous DE Solution

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Initial Value Problems (IVPs)

Initial Value Problem (Basic)

Solve the initial value problem:

dydx=2xy,y(0)=3\frac{dy}{dx} = 2xy, \quad y(0) = 3

Basic IVP Solution

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Initial Value Problem (Advanced)

Solve the initial value problem:

(e2y−y)cos⁡(x)dydx=eysin⁡(2x),y(0)=0(e^{2y} - y)\cos(x) \frac{dy}{dx} = e^y \sin(2x), \quad y(0) = 0

Advanced IVP Solution

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