In This Lesson What Is a DE? Separable Equations First-Order Linear 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 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
Separable Equations
Form: dy/dx = g(x)·h(y)
Method: (1/h(y)) dy = g(x) dx → integrate both sides
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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 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 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 DE4GibMedQ8hobyGwlXlCh9vKh48wldhkGKGX7+IAsdykRYaMQSPOpJnVfOkUXEzcu+HHWc/WYIIUga25k/CCvNlmat4hrhrOItE+zhuShgT3wGG19D5QotT+ysh201nKCvzyUVWVeWeLnBky0N0WTzMiSaa8bVlY+1tT8EL9BaddSZNlsfCcPOtJefXJShHx8AJ6197WmtWXoOtqloJMonC3F5KpvSW3HNNbfysQaIR9iLXbcZaAniqBl3qzLSz9fBeTUVBtlq9hffu/adxmcHJHfDZak10UY/NdlLCTZ+NU0dZsTfUGSQHVcKahxh4NHrOKnwI1q1velIVk9sBrgU74I621BD4zr25Ap/8fljOWClyp1h4yGW4QWYHNKINGwylwpRkrGGxKTql6TpBhndi15feimaN2wr2pLPeKGxgeixmqYxrRwtcREaiIy7Uzl84P4BDjcBVw1RTrDcuuY1I6VcDwSFGu5ai8WR36wxHN6EIcVd5IGPc4bW+He5xmJ3vZTUmu2YMslKns3DCNILfhP+l+Nwdk94Xsjrxeo1RQjzUs34JijjmBylI2jY9UCN5kXkoLuIe4Abvn58ArDBzfFwWJ46iSSWc9cW+VUYJhqzTmOcbcJPpVGQ7JFJu9DE2OCu3kHD8cHJVv7SdUAJlA5/PEJ4pN1aptP6Y8jTYQkRyfHvJadnjphisNP+ynvPJuHAdXMmugpfYBVoycqGI9EimYt2CLwxoieXV4KDYdYbtW9h0JdsWu7RfNe/lkrQbVKqDAuO/Z5mfiARnAgxCiAqjtdlg1ttAaK4LCmk+SrapZeT73KR9ULpfoCGDIGFoBfh6TIKq57VgjAn/TSm37hXgR9+Qed2k5rFIGNAr3imxaMuDQDDkUKgHba2oRzk8nHs+qVwGBEMmbZgT2CWwKIp7aFXvewqJtjuAfZBFLh8lGhEpnF1aPqcv5dwBJpLElAg3bdEblaMQLaMWg2uQf5N 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).