Differential Equations and Modeling

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A differential equation is an equation involving derivative(s) of the dependent variable with respect to the independent variable(s). Example: dy/dx + y = x² (involves the derivative dy/dx of y with r

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The order of a differential equation is the highest ordered derivative involved. For d³y/dx³ + x²(d²y/dx²)³ = 0, the highest derivative is d³y/dx³ (third derivative), so the order is 3.

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The degree is the highest power of the highest order derivative when the equation is polynomial in derivatives. For (d²y/dx²)² - 3(dy/dx)⁴ = y², the highest order derivative is d²y/dx² raised to power

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The degree is not defined when the differential equation is not polynomial in its derivatives. Example: d²y/dx² + y² + e^(dy/dx) = 0 (contains e^(dy/dx), which is not polynomial in dy/dx).

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General solution: Contains arbitrary constants and represents a family of curves. Example: y = ce^(-x) + 1. Particular solution: No arbitrary constants, obtained by giving specific values to constants

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Given: y = ae^(2x) Differentiate: dy/dx = 2ae^(2x) Substitute in equation: dy/dx - 2y = 2ae^(2x) - 2(ae^(2x)) = 2ae^(2x) - 2ae^(2x) = 0 ✓ Therefore, y = ae^(2x) is indeed a solution.

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Steps: 1) Start with equation f(x,y,a₁,a₂,...,aₙ) = 0, 2) Differentiate n times to get n additional equations, 3) Eliminate all n parameters from the (n+1) equations to get the differential equation.

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Given: x² + y² = a² ... (1) Differentiate: 2x + 2y(dy/dx) = 0 Simplify: x + y(dy/dx) = 0 This gives: x + y(dy/dx) = 0 or dy/dx = -x/y This is the required differential equation.