Differential Equations Intro
Grade 12 · Math · Study Guide
Understand what a differential equation is and how to solve separable first-order equations.
A differential equation (DE) is an equation involving a function and its derivatives, such as dy/dx = ky. A solution is a function y = f(x) that satisfies the equation. The general solution contains an arbitrary constant C; an initial condition determines the particular solution.
Separable equations are solved by algebraically separating variables: move all y-terms to one side and all x-terms to the other, then integrate both sides. For example, dy/dx = ky separates to dy/y = k dx, integrating to ln|y| = kx + C, giving y = Ae^(kx).
A slope field (direction field) is a graphical tool that shows the slope dy/dx at each point (x, y), giving a visual picture of all possible solution curves without solving the equation analytically.
| Step | Action |
|---|---|
| 1 | Separate variables: g(y) dy = f(x) dx |
| 2 | Integrate both sides |
| 3 | Solve for y (general solution) |
| 4 | Apply initial condition to find C |
| Model | Equation | Solution |
|---|---|---|
| Exponential growth | dy/dt = ky, k > 0 | y = y₀ e^(kt) |
| Exponential decay | dy/dt = ky, k < 0 | y = y₀ e^(kt) |
| Newton's cooling | dT/dt = k(T - Tₑ) | T = Tₑ + (T₀ - Tₑ)e^(kt) |
Key terms
- Differential equation
- An equation relating a function to one or more of its derivatives.
- Separable equation
- A DE where variables can be separated: g(y)dy = f(x)dx.
- General solution
- The family of all solutions, including the arbitrary constant C.
- Slope field
- A plot of short line segments showing dy/dx at each (x, y) point.
Practice questions
Which of the following is a differential equation?
- y = 3x + 2
- dy/dx = 2y
- ∫ y dx = x²
- y² + x = 5
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A differential equation relates a function to its derivative. dy/dx = 2y fits that definition.
Solve the separable DE dy/dx = 3x², given y(0) = 1.
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dy = 3x² dx → y = x³ + C. y(0)=1: 1 = 0 + C → C = 1. So y = x³ + 1.
The general solution to dy/dx = -2y is:
- y = Ce^(2x)
- y = Ce^(-2x)
- y = -2x + C
- y = Ce^x
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Separate: dy/y = -2 dx → ln|y| = -2x + C₀ → y = Ce^(-2x).
An initial condition is needed to find the particular solution of a differential equation.
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The general solution has arbitrary constant C. An initial condition like y(0) = 5 pins down C to give the particular solution.
A population grows according to dP/dt = 0.05P. If P(0) = 200, find P(t).
Show answer
Exponential growth model: P = P₀ e^(kt). P₀ = 200, k = 0.05. So P(t) = 200e^(0.05t).
Fun facts
- Differential equations model nearly everything in physics — Newton's laws, Maxwell's electromagnetic equations, and Schrödinger's quantum mechanics are all differential equations.
- Exponential growth models like dy/dt = ky describe population growth, compound interest, and radioactive decay all with the same equation.
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