In This Lesson What Is a DE? Separable Equations 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 First-Order Linear Exact Equations Applications 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
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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
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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 Mixing problems: Rate in − rate out → first-order linear DE Logistic growth: dP/dt = rP(1 − P/K) → S-shaped curve — uses partial fractions to integrate
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).
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