Unit 7: Differential Equations
Separable Differential Equations
Sketching slope fields and solving separable differential equations.
Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed
Visualizing and Solving an Equation Written in Terms of a Slope
Definition — Slope Field and Separable Equation
A slope field draws the local slope dy/dx = f(x,y) at each point of a grid, without solving the equation — a solution curve stays tangent to the field everywhere it passes. An equation is separable when its y-terms (with dy) and x-terms (with dx) can be moved to opposite sides and integrated separately, then solved for y using any given initial condition.
Worked Example — Reading a Slope Field, Then Solving the Same Equation Exactly
For dy/dx = x/y, a few slope values: at (1,1) the slope is 1; at (2,1) it's 2; at (1,2) it's 0.5; at (2,2) it's 1 — steeper farther from the line y = x, flatter near it. Now solve the same equation given y(3) = 4. Separate: y dy = x dx. Integrate: y²/2 = x²/2 + C, so y² − x² = K. Apply the initial condition: 16 − 9 = 7 = K. So y² = x² + 7, and taking the positive root (since y(3) = 4 > 0), y = √(x²+7). Check: dy/dx = x/√(x²+7) = x/y — matching the original equation exactly.
Tip
Apply an initial condition only after fully solving for y — it's far simpler to substitute into y = (solved expression) than to track it through the integration steps themselves.
Common Mistakes
Leaving an x term mixed in with the y-side (or vice versa) when separating variables.
Every term must end up purely in x (with dx) or purely in y (with dy) before integrating.
Key Takeaways
- A slope field visualizes a differential equation's solutions without solving anything algebraically.
- A separable equation is solved by moving all y-terms to one side, all x-terms to the other, integrating both, then applying an initial condition.
Summary
The next lesson solves a differential equation unique to BC's curriculum — one that builds in its own growth limit.
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