In This Lesson What Is a DE? Separable Equations First-Order Linear Exact Equations Applications 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 What Is a Differential Equation? A DE is an equation involving a function and its derivatives . The order is the highest derivative present. A first-order DE has the form dy/dx = f(x, y).
General form: F(x, y, y') = 0
Solution: a function y = φ(x) satisfying the equation
General solution: family of solutions with arbitrary constant C
Particular solution: satisfies an initial condition y(x₀) = y₀
Separable Equations
Form: dy/dx = g(x)·h(y)
Method: (1/h(y)) dy = g(x) dx → integrate both sides
First-Order Linear
Form: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^(∫P(x)dx)
Solution: y = (1/μ) ∫ μ·Q dx
The integrating factor technique converts a non-separable DE into an exact derivative. This method uses the product rule in reverse and relies on integration techniques .
Exact Equations
Form: M(x,y)dx + N(x,y)dy = 0
Exact if: ∂M/∂y = ∂N/∂x
Solution: find F where ∂F/∂x = M and ∂F/∂y = N; then F(x,y) = C
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 Exactness connects to partial derivatives and conservative vector fields. Non-exact equations can sometimes be made exact with an integrating factor.
Applications Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ (population, radioactive decay) Newton's cooling: dT/dt = −k(T − Tₐ) → T = Tₐ + (T₀ − Tₐ)e⁻ᵏᵗ Mixing problems: Rate in − rate out → first-order linear 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 Logistic growth: dP/dt = rP(1 − P/K) → S-shaped curve — uses partial fractions to integrate 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
First-order DEs appear in every scientific field. In
systems of DEs , x' = Ax, eigenvalues determine whether solutions grow, decay, or oscillate. In
probability , the exponential distribution's memoryless property comes from the DE: f'(t) = −λf(t).